Group (mathematics) and San Francisco International Airport: Difference between pages

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Revision as of 05:21, 29 June 2007

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{{coor title d|37.614433|N|122.39132|W}}

[[Image:Clock_group.svg|thumb|right|300px|This picture illustrates how the hours in a clock form a group.]]

{{Airport frame}}

In [[abstract algebra]], a '''group''' is a [[set]] with a [[binary operation]] that satisfies certain axioms, detailed below. For example, the set of integers with addition is a group. The branch of mathematics which studies groups is called [[group theory]].

{{Airport title|name=San Francisco International Airport}}

[[Image:Sfologo.75.svg|75px]]

{{Airport image|airport_image=SFInternational.JPG}}

{{Airport infobox

| IATA = SFO

| ICAO = KSFO - [[Location identifier|FAA]]: SFO

| type = Public

| run by = San Francisco Airports Commission

| closest town = San Francisco, California

| elevation_ft = 13

| elevation_m = 3.96

| coordinates = {{coor dms|37|37|08|N|122|22|30|W|type:airport}}

}}

{{Runway title}}

{{Runway

| runway_angle = 10L/28R

| runway_length_f = 11,870

| runway_length_m = 3,618

| runway_surface = [[Asphalt]]

}}

{{Runway

| runway_angle = 10R/28L

| runway_length_f = 10,602

| runway_length_m = 3,231

| runway_surface = Asphalt

}}

{{Runway

| runway_angle = 1R/19L

| runway_length_f = 8,648

| runway_length_m = 2,636

| runway_surface = Asphalt

}}

{{Runway

| runway_angle = 1L/19R

| runway_length_f = 7,500

| runway_length_m = 2,286

| runway_surface = Asphalt

}}

{{Airport end frame}}

[[Image:SFO map.png|thumb|right|FAA diagram of SFO]]

{{redirect|SFO}}

{{for|the television series|San Francisco International Airport (TV series)}}

Many of the structures investigated in mathematics turn out to be groups. These include familiar number systems, such as the [[integer]]s, the [[rational number]]s, the [[real number]]s, and the [[complex number]]s under addition, as well as the non-zero rationals, reals, and complex numbers, under multiplication. Other important examples are the group of non-singular [[matrix (mathematics)|matrices]] under multiplication and the group of [[Inverse function|invertible functions]] under [[Function composition|composition]]. Group theory allows for the properties of such structures to be investigated in a general setting.

'''San Francisco International Airport''' {{Airport codes|SFO|KSFO|SFO}} is a major international [[airport]] located 13 [[mile]]s (21 [[Kilometre|km]]) south of downtown [[San Francisco, California|San Francisco]], [[California]], [[United States]], adjacent to the cities of [[Millbrae, California|Millbrae]] and [[San Bruno, California|San Bruno]] in [[unincorporated area|unincorporated]] [[San Mateo County, California|San Mateo County]]. The airport has flights to destinations throughout the Americas and is a major gateway to [[Europe]], [[Asia]], and [[Australia]].

Group theory has extensive applications in mathematics, science, and engineering. Many [[algebraic structure]]s such as [[field (algebra)|field]]s and [[vector space]]s may be defined concisely in terms of groups, and group theory provides an important tool for studying [[symmetry]], since the symmetries of any object form a group. Groups are thus essential abstractions in branches of physics involving [[symmetry]] principles, such as [[Theory of relativity|relativity]], [[quantum mechanics]], and [[particle physics]]. Furthermore, their ability to represent [[Geometry|geometric]] transformations finds applications in [[chemistry]], [[computer graphics]], and other fields.

San Francisco is the largest airport in the [[San Francisco Bay Area]], and is the second busiest airport in the state of [[California]] after [[Los Angeles International Airport]]. [[As of 2005]], San Francisco International Airport is the fourteenth largest in the [[United States]]<ref name="14th">[http://www.aci-na.org/asp/traffic.asp?art=215 North America's largest airports by number of passengers.] Retrieved on [[August 7]] [[2006]].</ref> and the [[World's busiest airports by passenger traffic|twenty-third largest]] airport in the world,<ref name="23rd">[http://www.aci.aero/cda/aci/display/main/aci_content.jsp?zn=aci&cp=1-5-54-55-2812_9_2__ World's largest airports by number of passengers.] Retrieved on [[August 7]] [[2006]].</ref> in terms of passengers. It is a major [[Airline hub|hub]] of [[United Airlines]], and will become the main hub of [[Virgin America]] when the airline begins operations in mid-2007.

== History ==

<ref name="virginAmericaDOT">{{cite web

{{main|Group theory}}

| last =

| first =

| authorlink =

| coauthors =

| year =

| url = http://www.virginamerica.com/informationdesk/news/story_template.php?article=36

| title = Tentative Approval to Fly

| format =

| work =

| publisher = Virgin America

| accessdate = March 20

| accessyear = 2007

}}</ref>

The airport enjoys a connection to an adjacent freeway, [[U.S. Route 101]], as well as having its own [[Bay Area Rapid Transit|Bay Area Rapid Transit (BART)]] [[San Francisco International Airport (BART station)|station]] adjoining one of its terminals. [[Interstate 380 (California)|Interstate 380]] intersects Highway 101 north of the airport, providing further connections to the region.

== Definitions ==

A group ''(G, *)'' is a [[set]] ''G'' with an [[operation]] * that satisfies the following four [[axioms]]:

SFO has numerous passenger amenities, including a wide range of food and drink establishments, shopping, baggage storage, public showers, a medical clinic, and assistance for lost or stranded travelers and military personnel. The airport hosts the Louis A. Turpen Aviation Museum, the San Francisco Airport Commission Aviation Library, and both permanent and temporary art exhibitions in several places in the terminals. Public [[Wi-Fi]] is available throughout most of the terminal area, provided by [[T-Mobile]] for a fee.<ref name="WiFi">[http://www.flysfo.com/about/press/releases/SF-04-14.pdf Wi-Fi available in all areas of SFO.] Retrieved on [[August 7]] [[2006]].</ref>

* '' [[Closure (mathematics)|Closure]] '': * is a [[binary operation]]

* ''[[Associativity]]'': For all ''a'', ''b'' and ''c'' in ''G'', (''a'' * ''b'') * ''c'' = ''a'' * (''b'' * ''c'').

* ''[[Identity element]]'': There exists an element ''e'' in ''G'' such that for all ''a'' in ''G'', ''e'' * ''a'' = ''a'' * ''e'' = ''a''.

* ''[[Inverse element]]'': For each ''a'' in ''G'', there exists an element ''b'' in ''G'' such that ''a'' * ''b'' = ''b'' * ''a'' = ''e'', where ''e'' is an identity element.

==History==

By the definition of a binary operation, the group is [[Closure (mathematics)|closed]] under its operation, that is, for each [[ordered pair]] (a,b) in ''G'', the result of ''a'' * ''b'' is also in ''G''.

The airport was first opened on [[May 7]], [[1927]] on 150 acres (607,000 m²) of cow pasture. The land was leased from prominent local landowner [[Ogden L. Mills]], and was named Mills Field Municipal Airport. It remained Mills Field until [[1931]], when it was renamed San Francisco Municipal Airport. "Municipal" was replaced by "International" in [[1955]].

The [[U.S. Weather Bureau]] began keeping weather observations at Mills Field in May 1927. The weather records have continued under the [[National Weather Service]], which maintained the Bay Area forecast office in the airport's control tower building until forecasting was moved to [[Redwood City]]. Although not the official weather observation site for San Francisco (with the official site existing in [[Duboce Park]] in San Francisco's [[Mission District, San Francisco, California|Mission District]]), data from SFO's automated weather station often appears as belonging to "San Francisco" in media sources outside of the Bay Area.

Using the identity element property it can be shown that a group has exactly one identity element. See section simple theorems.

Starting in [[1935]], [[Pan American World Airways]] used the facility as the terminal for its "[[China Clipper]]" [[flying boat]] service across the [[Pacific Ocean]]. Domestic flights did not begin ''en masse'', however, until [[World War II]], when [[Oakland International Airport]] was taken over by the military and its passenger flights were shifted to San Francisco.<ref name="militaryOAK">[http://www.associatedcontent.com/article/9829/a_brief_history_of_oakland_international.html History of Oakland International Airport.] Retrieved on [[August 17]], [[2006]].</ref>

The inverse of an element can also be shown to be unique, and the left- and right-inverses of an element are the same. Some definitions are thus slightly more narrow, substituting the second and third axioms with the concept of a "left (or right) identity element" and a "left (or right) inverse element."

After the war, [[United Airlines]] took up residence at SFO, using the Pan Am terminal for its flights to [[Hawaii]] and other U.S. cities. In [[1954]], the airport's Central Passenger Terminal opened for passenger service.<ref name="intlterminalopen">[http://www.sfgate.com/cgi-bin/article.cgi?f=/c/a/2000/12/04/MN147657.DTL Unveiling of the new International Terminal.] From the ''San Francisco Chronicle''. Retrieved on [[August 7]], [[2006]].</ref> Jet service to SFO began in the late [[1950s]]: United built a large maintenance facility at San Francisco for its new [[Douglas DC-8]]s. In [[July]] [[1959]] the first [[jetway|jetway bridge]] was installed in the [[United States]]. In [[1974]], a new terminal was built for domestic flights, and the CPT became an international terminal (known today as Terminal 2).

Also note that a group ''(G,*)'' is often denoted simply ''G'' where there is no ambiguity in what the operation is.

===Operations, expansion, retreat and recovery===

== Basic concepts in group theory ==

In [[1989]], an airport master plan and associated [[Environmental Impact Report]] was prepared to guide expansion and development over the next two decades.<ref>''Environmental Impact Report for the San Francisco International Airport Master Plan'', Earth Metrics Inc. and Jefferson Associates, prepared for the city of San Fracisco and California State Clearinghouse (1989)</ref>

=== Order of groups and elements ===

During the economic boom of the 1990s and the [[dot-com bubble|dot-com boom]], SFO became the sixth busiest international airport in the world. However, since [[2001]], when the economic boom times ended, SFO has fallen back out of the top twenty.<ref name="23rd"/>

The '''order of a group ''G''''', denoted by |''G''|, is the number of elements in the set ''G''. If the order is not finite, then the group is an ''infinite group'', denoted |''G''|&nbsp;=&nbsp;∞.

SFO has expanded continuously through the decades. Most recently, a new $1 billion international terminal opened in [[December]] [[2000]], replacing Terminal 2 as the international terminal.<ref name="intlterminalopen"/> This new terminal contains a world-class aviation library and museum.<ref name="sfoarts">[http://www.sfoarts.org/ San Francisco Airport Museum.] Retrieved on [[August 7]], [[2006]].</ref> A long-planned extension of the [[Bay Area Rapid Transit]] system to the airport opened on [[June 22]] [[2003]], allowing passengers to board trains directly at the airport's international terminal bound for San Francisco or points in the East Bay.<ref name="BARTtoSFOstart">[http://www.sfgate.com/cgi-bin/article.cgi?file=/c/a/2003/06/22/MN100396.DTL BART to SFO service begins.] From the ''San Francisco Chronicle''. Retrieved on [[August 7]], [[2006]].</ref> BART trains also offer a quick trip to the nearby [[Millbrae Station]], where passengers can board [[Caltrain]] [[commuter rail]] trains bound for [[San Jose, California|San Jose]] and the [[San Francisco Peninsula|Peninsula]] and [[SamTrans]] [[bus]] service bound for the Peninsula. In [[2003]], the [[AirTrain (SFO)|AirTrain]] shuttle system opened, conveying passengers between terminals, parking lots, the BART station, and the rental car center on small automatic trains.

The '''order of an element ''a''''' in a group ''G'' is the least positive integer ''n'' such that ''aⁿ&nbsp;=&nbsp;e'', where ''aⁿ'' is multiplication of ''a'' by itself ''n'' times (or other suitable composition depending on the group operator). If no such ''n'' exists, then the order of ''a'' is said to be infinity.

[[Image:SFO at night.jpg|left|250px|thumb|San Francisco International Airport at night]]

===Subgroups===

It is not uncommon for SFO to experience significant delays in adverse weather, when only one of the airport's four runways can be used a time, due to a lateral separation of only 750 feet between runways. Airport planners have floated proposals to extend the airport's runways further into [[San Francisco Bay]] in order to accommodate the next generation of super-jumbo aircraft. In order to expand further into the bay, the airport would be required by law to restore bay land elsewhere in the Bay Area to offset the fill. Such proposals have nevertheless met resistance with environmental groups, fearing damage to the habitat of animals living near the airport and bay water quality.

A set ''H'' is a '''[[subgroup]]''' of a group ''G'' if it is a subset of ''G'' and a group using the operation defined on ''G''. In other words, ''H'' is a subgroup of (''G'', *) if the restriction of * to ''H'' is a group operation on ''H''.

As such, SFO suffers from loss of service as many airlines, especially as [[low-cost carrier]]s such as [[ATA Airlines]] increasingly shift service to the other two major Bay Area airports at [[Oakland International Airport|Oakland]] and [[San Jose International Airport|San Jose]], which continue to expand for the time being. However, SFO has superior land connections compared to Oakland and San Jose, being directly connected to [[U.S. Route 101]], [[Interstate 380 (California)|Interstate 380]], and the [[Bay Area Rapid Transit|BART system]].

If ''G'' is a finite group, then so is ''H''. Further, the order of ''H'' [[divisor|divides]] the order of ''G'' ([[Lagrange's theorem (group theory)|Lagrange's Theorem]]).

However, recovery at SFO has been evident. [[Spirit Airlines]] began daily service to [[Detroit]] on [[May 25]], [[2006]]. In addition, [[Qantas]] began service from [[Sydney]] in [[March]] [[2006]], and began service to [[Vancouver]] on [[June 14]], [[2006]]. United Airlines reinstated non-stop service to [[Taipei]] on [[June 7]] [[2007]].<ref name="SFO2AsiaPacficxpand">[http://www.united.com/press/detail/0,6862,54554,00.html United Airlines Boosts Asia-Pacific Service.] Retrieved on [[August 7],] [[2006]].</ref> In addition, SFO will be the base of operations for [[Virgin America]] when the airline begins operations in August [[2007]].

===Abelian groups===

A group ''G'' is said to be an '''[[abelian group]]''' (or '''commutative''') if the operation is commutative, that is, for all ''a'', ''b'' in ''G'', ''a'' * ''b'' = ''b'' * ''a''. A '''non-abelian''' group is a group that is not abelian. The term "abelian" is named after the mathematician [[Niels Abel]].

During the beginning of the summer season in [[2006]], low-cost carrier [[Frontier Airlines]] began operating flights to [[Los Angeles International Airport|Los Angeles]] adding on to its existing service to [[Denver International Airport|Denver]], Following the additional service to Los Angeles. Frontier began operating flights between SFO and [[McCarran International Airport|Las Vegas]] on [[December 14]], [[2006]]. These flights however have been scheduled to end on [[July 10]].

===Cyclic groups===

A '''[[cyclic group]]''' is a group whose elements may be [[Generating set of a group|generated]] by successive [[function composition|composition]] of the operation defining the group applied to a single element of that group. This single element is called the generator or [[Primitive root modulo n|primitive element]] of the group. For instance, in the case of a cyclic [[multiplicative group]] ''G'', all of the elements ''aⁿ'' of the group are generated by the set of all integer multiplications of a primitive element of that group (in [[set-builder notation]]):

On [[January 9]], [[2007]], [[JetBlue Airways]] announced they will begin five non-stop flights to New York's JFK and Boston's Logan airports starting May 3.<ref>[http://www.sfgate.com/cgi-bin/article.cgi?f=/c/a/2007/01/09/BUGJSNF21D1.DTL&hw=jetblue&sn=002&sc=864 JetBlue announces SFO flights] Retrieved on [[February 9]], [[2007]]</ref> On [[February 9]], [[2007]], [[Southwest Airlines]] announced their plans to resume serving San Francisco International Airport in the early fall of 2007,<ref>[http://www.southwest.com/about_swa/press/070209_san_francisco.html?ref=sfo_press_070208&type=b Southwest Airlines announces intent to resume service at SFO.] Retrieved on [[February 9]], [[2007]].</ref> after having pulled out of the airport in May 2001 citing high costs and delays. Irish airline [[Aer Lingus]] announced commencing service to [[Dublin, Ireland]] beginning [[October 28]] [[2007]] following the passage of the open skies treaty.

<math>G = \{ a^n \mid n \in \Z \pmod{m \in \Z} \}</math>

A global warming study unveiled in February 2007 revealed that much of SFO would be under water with only a one-meter rise in sea levels.<ref>http://www.sfgate.com/cgi-bin/article.cgi?f=/c/a/2007/02/18/MNG6SO72DJ1.DTL</ref>

For example if ''a'' is 2 and the operation is the mathematical multiplication operator, then ''G'' = <math>\{ 2^0, 2^1, 2^2, 2^3, .. \}</math> = <math>\{ 1, 2, 4, 8, .. \}</math>. The [[Modular arithmetic|modulo]] ''m'' may bind the group into a [[finite set]].

In April 2007, SFO announced plans to introduce a registered traveler program that would allow travelers to speed through the TSA security checkpoint in about 30 seconds.<ref>http://www.examiner.com/a-669843~Speedy_entry_coming_to_SFO.html</ref>

If successive composition of the operation defining the group is applied to a non-primitive element of the group, a [[cyclic subgroup]] is generated whose order divides the order of the group. Thus, if the order of a group is prime, all of its elements, except the identity, are primitive elements of the group. It is important to note that a group contains all the cyclic [[subgroups]] generated by each of the elements of ''G''. However, a group constructed from cyclic subgroups is itself not necessarily a cyclic group, e.g., a [[Klein four-group|Klein group]] is not a cyclic group even though it is constructed from two copies of the cyclic group of order 2.

Passenger and checked baggage screening is done by [[Covenant Aviation Security]] [http://www.cassfocustomerservice.com/], a [[Transportation Security Administration|TSA]] contractor. They are nicknamed "Team SFO."

== Notation for groups ==

== Aircraft noise abatement ==

Groups can use different notation depending on the context and the group operation.

[[Image:SFOApril2005.JPG|right|250px|thumb|San Francisco International Airport in the last rays of an April day]]

* Additive groups use ''+'' to denote addition, and the minus sign ''-'' to denote inverses. For example, ''a + (-a) = 0'' in ''Z''.

SFO was one of the first airports to implement a Fly Quiet Program which grades individual air carriers on their performance on noise abatement procedures while flying in and out of SFO. The [http://www.flyquietsfo.com/FlyQuiet.htm Jon C. Long Fly Quiet Program] is an initiative implemented by the Aircraft Noise Abatement Office to encourage individual airlines to operate as quietly as possible at SFO. The program promotes a participatory approach in complying with the noise abatement procedures.

* Multiplicative groups use ''*'' to denote multiplication, and the superscript ''^-1'' to denote inverses. For example, ''a * a^-1 = 1''. It is very common to drop the ''*'' and just write ''aa^-1'' instead.

* Function groups use ''•'' to denote function composition, and the superscript ''^-1'' to denote inverses. For example, ''g • g^-1 = e''. It is very common to drop the ''•'' and just write ''gg^-1'' instead.

SFO was also one of the first U.S. airports to conduct a residential sound abatement retrofitting program. Established by the [[Federal Aviation Administration|FAA]] in the early [[1980s]], this program evaluated the cost effectiveness of reducing interior sound levels for homes in the vicinity of the airport, or more particularly homes within the 65 [[Ambient noise level|CNEL]] noise contour surface. The program made use of a [[noise pollution|noise]] [[computer model]] to predict improvement in specific residential interiors for a variety of different [[noise mitigation|noise control]] strategies. This pilot program was conducted for a neighborhood in the city of [[South San Francisco, California|South San Francisco]], and success was achieved in all of the homes analyzed. The construction costs turned out to be modest, and the post-construction interior sound level tests confirmed the model predictions for noise abatement. To date over $137 million has been spent to insulate in excess of 15,000 homes throughout the neighboring cities of [[Daly City, California|Daly City]], [[Pacifica, California|Pacifica]], [[San Bruno, California|San Bruno]], and South San Francisco.<ref name="SFOnapstats">[http://www.flyquietsfo.com/ResSoundInsulation.htm Residential Sound Insulation Program.] Retrieved on [[August 7]] [[2006]].</ref>

Omitting a symbol for an operation is generally acceptable, and leaves it to the reader to know the context and the group operation.

{{seealso|Noise mitigation#Aircraft noise abatement|Aircraft noise}}

== Terminals, airlines and destinations ==

When defining groups, it is [[standard notation]] to use parentheses in defining the group and its operation. For example, (H,+) denotes that the set H is a group under addition. For groups like (Z<sub>n</sup>,+) and (F_n*, *) it is common to drop the parentheses and the operation, e.g. Z<sub>n</sup> and F_n*. It is also correct to refer to a group by its set identifier, e.g. ''H'' or '''Z'''.

<!--Airport map needed-->

The airport is composed of four [[Airport terminal|terminals]], in which two (Terminals 1 and 3) are domestic, one is international, and the fourth (Terminal 2) is under renovation. Within the framework of the terminals, the airport is split into seven concourses, in which four (Boarding Areas B, C, E, and F) are domestic, two (Boarding Areas A and G) are international, and one (Boarding Area D) is unused. Originally named the South, Central, and North Terminals, the domestic terminals were renamed Terminals 1, 2, and 3, respectively, after the new international terminal opened.

''Note:'' All flights to [[Canada]] depart from the domestic terminals, and [[JetBlue Airways]] and [[Spirit Airlines]] depart from International Terminal Boarding Area A.

The identity element ''e'' is sometimes known as the "neutral element," and is sometimes denoted by some other symbol, depending on the group:

* In multiplicative groups, the identity element can be denoted by 1.

* In invertible matrix groups, the identity element is usually denoted by I.

* In additive groups, the identity element may be denoted by 0.

* In function groups, the identity element is usually denoted by f₀.

===Terminal 1===

If ''S'' is a subset of ''G'' and ''x'' an element of ''G'', then, in multiplicative notation, ''xS'' is the set of all products {''xs'' : ''s'' in ''S''}; similarly the notation ''Sx'' = {''sx'' : ''s'' in ''S''}; and for two subsets ''S'' and ''T'' of ''G'', we write ''ST'' for {''st'' : ''s'' in ''S'', ''t'' in ''T''}. In additive notation, we write ''x'' + ''S'', ''S'' + ''x'', and ''S'' + ''T'' for the respective sets (see [[cosets]]).

[[Image:SFO Terminal 1.png|thumb|150px|Terminal 1]]

Formerly known as the ''South Terminal'', Terminal 1 consists of ''Boarding Area B'' and ''Boarding Area C''.

====Boarding Area B (Gates 20-36)====

== Examples of groups ==

''Note:'' All Alaska Airlines domestic and Canadian flights depart and arrive at Terminal 1 Boarding Area B and all Alaska Airlines Mexican flights depart and arrive at International Terminal Boarding Area A.

: ''Main articles: [[Examples of groups]], [[List of small groups]]''

* [[Air Canada]] (Calgary, Montréal, Toronto-Pearson, Vancouver)

* [[AirTran Airways]] (Atlanta, Indianapolis)

* [[Alaska Airlines]] (Anchorage [seasonal], Los Angeles, Palm Springs, Portland (OR), San Diego [ends September 30], Seattle/Tacoma, Vancouver)

** [[Horizon Air]] (Portland (OR))

* [[Continental Airlines]] (Cleveland, Houston-Intercontinental, Newark)

* [[Southwest Airlines]] (Chicago-Midway, Las Vegas, San Diego) [begins August 26]

* [[Sun Country Airlines]] (Minneapolis/St. Paul)

* [[US Airways]] (Charlotte, Philadelphia, Pittsburgh)

** [[US Airways]] operated by [[America West Airlines]] (Las Vegas, Philadelphia, Phoenix)

** [[US Airways Express]] operated by [[Mesa Airlines]] (Phoenix)

====Boarding Area C (Gates 40-48)====

=== An abelian group: the integers under addition ===

[[Image:Sfo-areac.jpg|thumb|200px|right|Interior of Boarding Area C]]

''Note'': All Northwest Airlines domestic flights depart from Terminal 1 Boarding Area C and all Northwest international flights depart and arrive at International Terminal Boarding Area A.

* [[Delta Air Lines]] (Atlanta, Cincinnati/Northern Kentucky, Honolulu [ends September 4], New York-JFK, Salt Lake City)

** [[Delta Connection]] operated by [[ExpressJet Airlines]] (Los Angeles)

** [[Delta Connection]] operated by [[SkyWest]] (Salt Lake City)

* [[Frontier Airlines]] (Denver, Las Vegas [ends July 10], Los Angeles [ends July 10], Los Cabos [ends July 10])

** [[Frontier Airlines]] operated by [[Republic Airlines]] (Denver, Los Angeles [ends July 10])

* [[Hawaiian Airlines]] (Honolulu)

* [[Northwest Airlines]] (Detroit, Honolulu, Indianapolis [seasonal], Memphis [seasonal], Minneapolis/St. Paul)

===Terminal 2===

A familiar group is the group of [[integer]]s under [[addition]]. Let '''Z''' be the set of integers, {..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...}, and let the symbol "+" indicate the operation of addition. Then ('''Z''',+) is a group.

<!--Picture needed for Terminal 2 soon-->

Formerly known as the ''Central Terminal,'' in 1974 it became known as the ''International Terminal.'' Terminal 2 consists of ''Boarding Area D'', which formerly included gates 50-59. However, when the current international terminal opened in 2000, Terminal 2 was closed; it is currently undergoing indefinite renovation and serves as a walkway between Terminal 1 and Terminal 3. The terminal will replace Rotunda A once renovation is complete. The [http://www.flysfo.com/guide_nonflash/airportinfo/medical.asp SFO Medical Clinic] is located on the Arrivals/Baggage Claim level (lower level).

====Boarding Area D====

Proof:

''Closed pending renovation or reconstruction'' - circular lounges are in the process of being demolished in late 2006 and early [[2007]]

* '''Closure''': If ''a'' and ''b'' are integers then ''a'' + ''b'' is an integer.

* '''Associativity''': If ''a'', ''b'', and ''c'' are integers, then (''a'' + ''b'') + ''c'' = ''a'' + (''b'' + ''c'').

* '''Identity element''': 0 is an integer and for any integer ''a'', 0 + ''a'' = ''a'' + 0 = ''a''.

* '''Inverse elements''': If ''a'' is an integer, then the integer −''a'' satisfies the inverse rules: ''a'' + (−''a'') = (−''a'') + ''a'' = 0.

===Terminal 3===

This group is also abelian because ''a'' + ''b'' = ''b'' + ''a''.

[[Image:SFO Airport Terminal 3.png|thumb|Terminal 3]]

Formerly known as the ''North Terminal,'' Terminal 3 is made up of ''Boarding Area E'' and ''Boarding Area F''. This terminal is utilized by [[United Airlines]], [[Midwest Airlines]] and [[American Airlines]], chiefly by United.

====Boarding Area E (Gates 60-67) - American Airlines====

If we extend this example further by considering the integers with both addition and multiplication, which forms a more complicated algebraic structure called a [[ring (algebra)|ring]]. (But, note that the integers with multiplications are ''not'' a group)

* [[American Airlines]] (Boston, Chicago-O'Hare, Dallas/Fort Worth, Honolulu, Los Angeles, Miami, New York-JFK, St. Louis)

** [[American Eagle Airlines|American Eagle]] (Orange County)

* [[Midwest Airlines]] (Kansas City, Milwaukee [seasonal])

====Boarding Area F (Gates 68-90) - United Airlines====

=== ''Not'' a group: the integers under multiplication ===

''Note:'' All United Airlines domestic and Canadian flights depart from Terminal 3 Boarding Area F and all United international flights depart and arrive at International Terminal Boarding Area G.

* [[United Airlines]] (Anchorage [seasonal], Atlanta, Austin, Baltimore/Washington, Boston, Burbank, Chicago-O'Hare, Dallas/Fort Worth, Denver, Honolulu, Houston-Intercontinental, Kahului, Kona, Lihue, Los Angeles, New York-JFK, Newark, Orange County, Orlando, Philadelphia, Portland (OR), Reno/Tahoe, Salt Lake City, San Antonio [begins September 11], San Diego, Seattle/Tacoma, Toronto-Pearson, Vail/Eagle [seasonal], Vancouver, Washington-Dulles)

** [[Ted (airline)|Ted]] operated by [[United Airlines]] (Las Vegas, Phoenix)

** [[United Express]] operated by [[SkyWest]] (Albuquerque, Aspen [seasonal], Bakersfield, Billings, Boise, Bozeman [seasonal], Burbank, Calgary, Chico, Colorado Springs, Crescent City, Edmonton, Eugene, Eureka, Fresno, Medford, Modesto, Monterey, Ontario, Orange County, Palmdale, Palm Springs, Redding, Redmond/Bend, Reno/Tahoe, Sacramento, Salt Lake City, San Antonio [ends September 10], San Luis Obispo, Santa Barbara, Tucson)

===International Terminal===

On the other hand, if we consider the integers with the operation of [[multiplication]], denoted by "·", then ('''Z''',·) is not a group. It satisfies most of the axioms, but fails to have inverses:

[[Image:SFO Airport International Terminal.png|thumb|255px|International Terminal]]

* '''Closure''': If ''a'' and ''b'' are integers then ''a'' · ''b'' is an integer.

[[Image:San Francisco International Airport International Terminal.jpg|thumb|255px|Exterior view of the International Terminal]]

* '''Associativity''': If ''a'', ''b'', and ''c'' are integers, then (''a'' · ''b'') · ''c'' = ''a'' · (''b'' · ''c'').

SFO's international terminal, which opened in [[December 2000]], is the largest international terminal in [[North America]], and is the largest building in the world built on [[Seismic retrofit|base isolators]] to protect against [[earthquake]]s.<ref name="intlterminalfactsheet">[http://www.flysfo.com/about/press/factsheets/International_Terminal_Fact_Sheet.pdf International Terminal Fact Sheet.] Retrieved on [[August 22]] [[2006]].</ref> It replaced Terminal 2, which served as SFO's international terminal until [[2000]]. The boarding areas have two levels, with shops and restaurants on the upper level and departure lounges on the lower level. Instead of the customary fast-food chains found at many other airports across the country, all restaurants in the International Terminal are leading restaurants in the Bay Area that have opened up fast-food versions of their establishments. SFO planners attempted to make the airport a destination in and of itself, not just for travelers that are passing through.<ref name="SFOGastromony">{{cite web

* '''Identity element''': 1 is an integer and for any integer ''a'', 1 · ''a'' = ''a'' · 1 = ''a''.

| last =

* However, it is '''not''' true that whenever ''a'' is an integer, there is an integer ''b'' such that ''ab'' = ''ba'' = 1. For example, ''a'' = 2 is an integer, but the only solution to the equation ''ab'' = 1 in this case is ''b = 1/2''. We cannot choose ''b'' = 1/2 because 1/2 is not an integer. (Inverse element ''fails'')

| first =

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| url = http://sfgate.com/cgi-bin/article.cgi?f=/c/a/2003/10/07/BUG9Q26KPD1.DTL&type=business

| title = Terminal gastronomy

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| publisher = ''San Francisco Chronicle''

| accessdate = October 7

| accessyear = 2003

}}</ref> The international terminal is a [[common use facility]], with all gates and all ticketing areas shared among the international airliners.

The [[San Francisco International Airport (BART station)|airport BART station]] is also located in this terminal, at the garage leading to Boarding Area G.

Since not every element of ('''Z''',·) has an inverse, ('''Z''',·) is ''not'' a group. It is, however, a commutative [[monoid]], which is a similar structure to a group but does not require inverse elements.

All the gates in this terminal have two [[Jetway|jetway bridges]] for use by [[Boeing 747]] aircraft, which are frequent visitors to the terminal, as it is a major transpacific gateway. Six of these gates are specifically designed for the Airbus A380, making SFO one of the first airports in the world with such gates when it was constructed in 2000.<ref>Armstrong, David. "[http://sfgate.com/cgi-bin/article.cgi?f=/c/a/2004/07/15/BUG7H7KVDS1.DTL&hw=A380&sn=003&sc=726 Super-size skies / SFO says it's ready for a 555-person plane arriving in 2006.]" ''[[San Francisco Chronicle]].'' [[July 15]], [[2004]]. Retrieved on [[September 12]], [[2006]].</ref>

=== An abelian group: the nonzero rational numbers under multiplication ===

For lack of space, the terminal was constructed ''on top'' of the airport's main access road at enormous expense; the advantage of this ___location was that it completed a continuous "ring" of terminals around the airport's main loading/unloading loop. The disadvantage was that the terminal required its own elaborate set of ramps to connect it with Highway 101.

Consider the set of [[rational number]]s '''Q''', the set of all fractions of integers ''a''/''b'', where ''a'' and ''b'' are integers and ''b'' is nonzero, and the operation multiplication, denoted by "·". Since the rational number [[0 (number)|0]] does not have a multiplicative inverse, ('''Q''',·), like ('''Z''',·), is not a group.

The design and construction of the international terminal is owed to [[Skidmore, Owings & Merrill]], Del Campo & Maru Architects, Michael Willis Associates (main terminal building), [[Hellmuth, Obata and Kassabaum]] (Boarding Area G) & Gerson/Overstreet Architects (Boarding Area A).<ref name="intlterminalfactsheet"/> The contracts were awarded after an architectural design competition.

However, if we instead use the set of all ''nonzero'' rational numbers '''Q''' \ {0}, then ('''Q''' \ {0},·) ''does'' form an abelian group.

* '''Closure''', '''Associativity''', and '''Identity element''' axioms are easy to check and follow because of the properties of integers.

* '''Inverse elements''': The inverse of ''a''/''b'' is ''b''/''a'' and it satisfies the axiom.

Despite the terminal's name, [[Spirit Airlines]] serves domestic destinations using this terminal, in Boarding Area A, primarily due to lack of available gates{{Fact|date=May 2007}} in the domestic terminals. [[JetBlue Airways]] operates in Boarding Area A. [[Virgin America]] also plans to use the international terminal for its operations.<ref>Armstrong, David. "[http://sfgate.com/cgi-bin/article.cgi?file=/c/a/2007/03/21/MNGRUOOVA71.DTL Startup Airline Revving up its Engines.]" ''[[San Francisco Chronicle]].'' [[March 21]], [[2007]]. Retrieved on [[March 21]], [[2007]].</ref><ref>Murtagh, Heather. "[http://www.smdailyjournal.com/article_preview.php?id=72463 Virgin America Cleared.]" ''[[San Mateo Daily Journal]].'' [[March 21]], [[2007]]. Retrieved on [[March 21]], [[2007]].</ref>

We don't lose closure by removing zero, because the product of two nonzero rationals is never zero. Just as the integers form a [[ring (mathematics)|ring]], the rational numbers form the algebraic structure of a [[field (mathematics)|field]], allowing the operations of addition, subtraction, multiplication and division.

When there are no gates available in one international boarding area, airlines will deplane from the other international boarding area.

=== A finite nonabelian group: permutations of a set ===

: ''This example is taken from the larger article on the [[Dihedral group of order 6]]''

For a more concrete example of a group, consider three colored blocks (red, green, and blue), initially placed in the order RGB. Let ''a'' be the action "swap the first block and the second block", and let ''b'' be the action "swap the second block and the third block".

====Boarding Area A (Gates A1-A12)====

[[Image:GroupDiagramD6.png|frame|right|[[Cycle diagram]] for S₃. A loop specifies a series of powers of any element connected to the identity element (1). For example, the e-ba-ab loop reflects the fact that (ba)²=ab and (ba)³=e, as well as the fact that (ab)²=ba and (ab)³=e The other "loops" are roots of unity so that, for example a²=e.]]

(south side, opposite Boarding Area G, next to Boarding Area B)

* [[Aer Lingus]] (Dublin) [begins October 28]

* [[Air France]] (Paris-Charles de Gaulle)

* [[Alaska Airlines]] (Cancún [seasonal], Ixtapa/Zihuatanejo, Los Cabos, Mazatlan, Puerto Vallarta)

* [[British Airways]] (London-Heathrow)

* [[Cathay Pacific]] (Hong Kong)

* [[China Airlines]] (Taipei-Taiwan Taoyuan)

* [[JetBlue Airways]] (Boston, New York-JFK, Salt Lake City [begins July 27])

* [[Japan Airlines]] (Tokyo-Narita)

* [[Jet Airways]] (Mumbai, Shanghai-Pudong) [begins October 2007]

* [[KLM|KLM Royal Dutch Airlines]] (Amsterdam)

* [[Korean Air]] (Seoul-Incheon)

* [[Mexicana de Aviación|Mexicana]] (Guadalajara, Mexico City, Morelia)

* [[Northwest Airlines]] (Bangkok-Suvarnabhumi, Tokyo-Narita)

* [[Philippine Airlines]] (Manila)

* [[Qantas]] (Sydney, Vancouver [seasonal])

* [[Spirit Airlines]] (Detroit [seasonal])

* [[Grupo TACA|TACA]] (San Salvador)

* [[Virgin Atlantic Airways|Virgin Atlantic]] (London-Heathrow)

====Boarding Area G (Gates G91-G102)====

In multiplicative form, we traditionally write ''xy'' for the combined action "first do ''y'', then do ''x''"; so that ''ab'' is the action RGB → RBG → BRG, i.e., "take the last block and move it to the front".

(north side, opposite Boarding Area A, next to Boarding Area F. All international [[Star Alliance]] members' flights use Boarding Area G.

If we write ''e'' for "leave the blocks as they are" (the identity action), then we can write the six [[permutation]]s of the [[set]] of three blocks as the following actions:

* [[Air China]] (Beijing)

* [[Air New Zealand]] (Auckland, Melbourne)

* [[All Nippon Airways]] (Tokyo-Narita)

* [[Asiana Airlines]] (Seoul-Incheon)

* [[EVA Air]] (Taipei-Taiwan Taoyuan)

* [[Lufthansa]] (Frankfurt, Munich)

* [[Singapore Airlines]] (Hong Kong, Seoul-Incheon, Singapore)

* [[United Airlines]] (Bangkok-Suvarnabhumi, Beijing, Cancun, Frankfurt, Ho Chi Minh City [ends October 28], Hong Kong, London-Heathrow, Mexico City, Nagoya-Centrair, Osaka-Kansai, Seoul-Incheon, Shanghai-Pudong, Sydney, Taipei-Taiwan Taoyuan, Tokyo-Narita)

** [[Ted (airline)|Ted]] operated for [[United Airlines]] (Los Cabos, Puerto Vallarta)

== Aircraft incidents ==

* ''e'' : RGB → RGB

* ''a'' : RGB → GRB

* ''b'' : RGB → RBG

* ''ab'' : RGB → BRG

* ''ba'' : RGB → GBR

* ''aba'' : RGB → BGR

On [[December 24]] [[1964]], [[Flying Tiger Line]] [[Flying Tiger Line Flight 282|Flight 282]], a cargo aircraft departing for [[New York City]], crashed in the hills west of the airport, killing all 3 crewmembers aboard.<ref name="FTL283">[http://ntsb.gov/ntsb/brief.asp?ev_id=77664&key=0 NTSB report on FTL 282.] From the [[NTSB]]. Retrieved on [[August 9]] [[2006]].</ref>

Note that the action ''aa'' has the effect RGB → GRB → RGB, leaving the blocks as they were; so we can write ''aa'' = ''e''.

Similarly,

* ''bb'' = ''e'',

* (''aba'')(''aba'') = ''e'', and

* (''ab'')(''ba'') = (''ba'')(''ab'') = ''e'';

so each of the above actions has an inverse.

On [[July 30]] [[1971]], [[Pan Am Flight 845]], a [[Boeing 747]] (registration: N747PA, name: Clipper America), struck navigational aids at the end of runway 1R on takeoff for [[Tokyo]]. The aircraft's landing gear was damaged, and the flight proceeded out over the [[Pacific Ocean]] to dump fuel in order to reduce weight for an emergency landing. Emergency services were deployed at the airport, and the plane returned and landed on runway 28R, using only the landing gear on one side of the aircraft. As the gear partially collapsed, the aircraft skidded into the dirt area next to the runway and came to a stop, but there was no fire. The aircraft was successfully evacuated using emergency slides. There were no fatalities among the 218 passengers and crew aboard, but there were a number of injuries, some serious. An investigation determined the cause of the accident to be erroneous information from the flight dispatcher to the crew regarding weight and runway length.<ref name="PAA845">[http://www.airdisaster.com/reports/ntsb/AAR72-17.pdf Airdisaster.com PDF report on PAA 845.] From the NTSB. Retrieved on [[August 9]] [[2006]].</ref>

By inspection, we can also determine associativity and closure; note for example that

* (''ab'')''a'' = ''a''(''ba'') = ''aba'', and

* (''ba'')''b'' = ''b''(''ab'') = ''bab''.

On [[February 19]], [[1985]], [[China Airlines Flight 006]], en route from [[Taipei]] to [[Los Angeles]], lost power over the Pacific in one of its four [[jet engine|engines]]. The pilots of the [[Boeing 747SP]] aircraft failed to trim the plane to counteract the asymmetric [[thrust]] condition, despite having several minutes to do so. The aircraft eventually rolled over and dived a total 30,000 feet before being brought under control and diverted to SFO.

This group is called the ''[[symmetric group]] on 3 letters'', or ''S''₃.

It has order 6 (or 3 [[factorial]]), and is non-abelian (since, for example, ''ab'' ≠ ''ba'').

Since ''S''₃ is built up from the basic actions ''a'' and ''b'', we say that the set {''a'',''b''} ''[[generating set of a group|generates]]'' it.

On [[June 28]], [[1998]], [[United Airlines Flight 863]] cleared nearby San Bruno Mountain by only 100 feet after a pilot erred in correcting for a failed engine during takeoff. <ref>[http://www.post-gazette.com/headlines/19990320jumbojet4.asp Post Gazette report.] Retrieved on [[April 17]] [[2007]].</ref>

More generally, we can define a [[symmetric group]] from all the permutations of ''N'' objects. This group is denoted by ''S''_''N'' and has order ''N'' factorial.

One of the flights during the [[September 11, 2001 attacks]], [[United Airlines Flight 93]], was bound for SFO.

One of the reasons that permutation groups are important is that every finite group can be expressed as a subgroup of a symmetric group ''S''_''N''; this result is [[Cayley's theorem]].

On [[May 26]], [[2007]] SkyWest Airlines flight 5741, an [[Embraer 120]], was involved in a serious runway incursion with Republic Airlines flight 4912, an [[Embraer 170]], on intersecting runways at [[San Francisco]].<ref>[http://ntsb.gov/ntsb/brief.asp?ev_id=20070610X00701&key=1 NTSB report]</ref>

=== Cyclic multiplicative groups ===

In the case of a cyclic [[multiplicative group]] ''G'', all of the elements ''aⁿ'' of the group are generated by the set of all integer [[exponentiation]]s of a primitive element of that group (in [[set-builder notation]]):

==In popular culture==

<math>G = \{ a^n \mid n \in \Z \pmod{m \in \Z} \}</math>

*The short-lived television series "[[San Francisco International Airport (TV series)|San Francisco International Airport]]" ([[1970]]) was set at the airport.

In this example if ''a'' is 2 and the operation is the mathematical multiplication operator, then ''G'' = <math>\{ 2^0, 2^1, 2^2, 2^3, .. \}</math> = <math>\{ 1, 2, 4, 8, .. \}</math>. The [[Modular arithmetic|modulo]] ''m'' may bind the group into a [[finite set]].

*The climax of the [[Steve McQueen]] movie "[[Bullitt]]" was filmed at the airport

==Trivia==

== Simple theorems ==

{{Trivia|date=June 2007}}

* A group has [[unique|exactly one]] identity element.

*Airlines that serve SFO lobbied heavily against extending the [[Bay Area Rapid Transit]] subway system into the airport complex. They preferred a lower cost alternative which would have placed the BART station approximately one mile away from the airport, but would have connected both BART and Caltrain to the airport via a westward extension of the [[AirTrain (SFO)|AirTrain]] people-mover system.

*Super short-haul routes, such as SFO-[[Oakland International Airport|OAK]] (11 miles) and SFO-[[San Jose International Airport|SJC]] (31 miles), were once served by [[United Airlines]].

*At 88 feet, SFO has one of the shortest [[control tower]]s of any major hub airport in the United States.{{Fact|date=April 2007}}

==See also==

:''Proof'': Suppose both ''e'' and ''f'' are identity elements. Then, by the definition of identity, ''fe'' = ''ef'' = ''e'' and also ''ef'' = ''fe'' = ''f''. But then ''e'' = ''f''.

*[[Oakland International Airport]]

*[[San Jose International Airport]]

:Therefore the identity element is unique.

*[[List of airports in the San Francisco Bay area]]

*[[Transportation in the San Francisco Bay Area]]

* Every element has exactly one inverse.

:''Proof'': Suppose both ''b'' and ''c'' are inverses of ''x''. Then, by the definition of an inverse, ''xb'' = ''bx'' = ''e'' and ''xc'' = ''cx'' = ''e''. But then:

::{|

|<math>xb = e = xc</math>

|-

|<math>xb = xc</math>

|-

|<math>bxb = bxc</math> || (multiplying on the left by ''b'')

|-

|<math>eb = ec</math> || (using ''bx'' = ''e'')

|-

|<math>b = c</math> || (neutral element axiom)

|}

:Therefore the inverse is unique.

The first two properties actually follow from associative binary operations defined on a set. Given a binary operation on a set, there is at most one identity and at most one inverse for any element.

* You can perform [[division (mathematics)|division]] in groups; that is, given elements ''a'' and ''b'' of the group ''G'', there is exactly one solution ''x'' in ''G'' to the [[equation]] ''x'' * ''a'' = ''b'' and exactly one solution ''y'' in ''G'' to the equation ''a'' * ''y'' = ''b''.

* The expression "''a''₁ * ''a''₂ * ··· * ''a''_''n''" is unambiguous, because the result will be the same no matter where we place parentheses.

* (''Socks and shoes'') The inverse of a product is the product of the inverses in the opposite order: (''a'' * ''b'')⁻¹ = ''b''⁻¹ * ''a''⁻¹.

:''Proof'': We will demonstrate that (ab)(b^-1a^-1) = (b^-1a^-1)(ab) = e, as required by the definition of an inverse.

:::{|

|<math>(ab)(b^{-1}a^{-1})</math> || = || <math>a(bb^{-1})a^{-1}</math> || (associativity)

|-

| || = || <math>aea^{-1}</math> || (definition of inverse)

|-

| || = || <math>aa^{-1}</math> || (definition of neutral element)

|-

| || = || <math>e</math> || (definition of inverse)

|}

:And similarly for the other direction.

These and other basic facts that hold for all individual groups form the field of [[elementary group theory]].

== Constructing new groups from given ones ==

Some possible ways to construct new groups from a set of given groups:

* '''[[Subgroup]]s''': A subgroup ''H'' of a group ''G'' is a group.

* '''[[Quotient group]]''': Given a group ''G'' and a [[normal subgroup]] ''N'', the quotient group is the set of [[coset]]s of ''G/N'' together with the operation (''gN'')(''hN'')=''ghN''.

* '''[[Direct product#Group direct product|Direct product]]''': If (''G'',*) and (''H'',•) are groups, then the set ''G''×''H'' together with the operation (''g''₁,''h''₁)(''g''₂,''h''₂) = (''g''₁*''g''₂,''h''₁•''h''₂). The direct product can also be defined with any number of terms, finite or infinite, by using the [[Cartesian product]] and defining the operation coordinate-wise.

* '''[[Semidirect product]]''': If ''N'' and ''H'' are groups and &phi; : ''H'' &rarr; Aut(''N'') is a [[group homomorphism]], then the semidirect product of ''N'' and ''H'' with respect to &phi; is the group (''N'' &times; ''H'', *), with * defined as

*: (''n''₁, ''h''₁) * (''n''₂, ''h''₂) = (''n''₁ &phi;(''h''₁) (''n''₂), ''h''₁ ''h''₂)

* '''[[Direct sum of groups|Direct external sum]]''': The direct external sum of a family of groups is the subgroup of the product constituted by elements that have a finite number of non-identity coordinates. If the family is finite the direct sum and the product are equivalent.

== Proving that a set is a group ==

There are two main methods in proving that a set is a group:

* Prove that the set is a [[subgroup]] of a group;

* Prove that the set is a group using the definition.

The first method is generally referred to as the "[[Subgroup Test]]" and requires that you prove the following if trying to prove that H is a subgroup:

* The set H is a [[non-empty subset]] of G (i.e. has the ''identity element'' inside)

* H is closed under the same operation as G. (ab is in H and a^-1 is in H for all a,b in H)

The second method requires that you prove all the axioms and assumptions in the definition for a set G:

* G is non-empty;

* G is closed under the binary operation;

* G is associative;

* e is in G (usually follows from non-emptiness);

* G consists of [[units]].

For [[finite]] groups, one only needs to prove that a subset is non-empty and is closed under the ambient group's operation.

==Generalizations==

In [[abstract algebra]], we get some related structures which are similar to groups by relaxing some of the axioms given at the top of the article.

* If we eliminate the requirement that every element have an inverse, then we get a [[monoid]].

* If we additionally do not require an identity either, then we get a [[semigroup]].

* Alternatively, if we relax the requirement that the operation be [[associative]] while still requiring the possibility of [[division (mathematics)|division]], then we get a [[loop (algebra)|loop]].

* If we additionally do not require an identity, then we get a [[quasigroup]].

* If we don't require any axioms of the binary operation at all, then we get a [[magma (algebra)|magma]].

[[Groupoid]]s, which are similar to groups except that the composition ''a'' * ''b'' need not be defined for all ''a'' and ''b'', arise in the study of more involved kinds of symmetries, often in topological and analytical structures.

They are special sorts of [[category theory|categories]].

[[supergroup (physics)|Supergroups]] and [[Hopf algebra]]s are other generalizations.

[[Lie group]]s, [[algebraic group]]s and [[topological group]]s are examples of [[group object]]s: group-like structures sitting in a [[category theory|category]] other than the ordinary category of sets.

Abelian groups form the prototype for the concept of an [[abelian category]], which has applications to [[vector space]]s and beyond.

[[Formal group law]]s are certain [[formal power series]] which have properties much like a group operation.

==References==

*[http://www.flysfo.com/ San Francisco International Airport] (official site)

*{{FAA-airport|ID=SFO|use=PU|own=PU|site=02187.*A}}

* Herstein, I.N. ''Abstract Algebra'', Wiley, ISBN 0-471-36879-2

===Notes===

* Dummit, David and Richard Foote. ''Abstract Algebra'', Wiley, ISBN 0-471-43334-9

<!--<nowiki>

* [[Serge Lang|Lang, Serge]]. ''Algebra'', Springer, ISBN 0-387-95385-X

See http://en.wikipedia.org/wiki/Wikipedia:Footnotes for an explanation of how

to generate footnotes using the <ref> and </ref> tags, and the template below

==See also==

</nowiki>-->

{{Wikibooks|Abstract algebra/Groups}}

<div class="references-small"><references/></div>

* [[Glossary of group theory]]

* [[Elementary group theory]]

* [[List of group theory topics]]

* [[List of publications in mathematics#Group theory|Important publications in group theory]]

* [[Examples of groups]]

* [[List of small groups]]

* [[Semidirect product]]

* [[Cosets]]

* [[Lagrange]]

* [[Galois]]

==External links==

{{commons|San Francisco International Airport}}

*[http://mathworld.wolfram.com/Group.html Group] at [[MathWorld]].

*[http://www.flysfo.com/ San Francisco International Airport website]

*[http://planetmath.org/encyclopedia/Group.html Group] at [[PlanetMath]].

*[http://www.smcroundtable.com/ San Francisco International Airport Community Roundtable Homepage]

*[http://www.flyquietsfo.com/ San Francisco International Airport Aircraft Noise Abatement Office]

*[http://www.protectourbay.com/level2/oversch.html Overscheduling at SFO]

*{{PDFlink|[http://www.flysfo.com/guide_nonflash/airportinfo/AirTrainBrochure.pdf AirTrain]|244&nbsp;[[Kibibyte|KiB]]<!-- application/pdf, 250188 bytes -->}}

*{{FAA-diagram|00375}}

{{US-airport|SFO}}

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