Group theory: Difference between revisions

Revision as of 03:34, 8 November 2003

Revision as of 18:32, 29 January 2004

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==History==

There are three historial roots of group theory: theory of algebraic equations, [[number theory]] and [[geometry]]. Euler, Gauss, Lagrange, Abel and [[Galois]] were precedent researchers in the field of group theory. Galois is honored as the first mathematician linking group theory with [[field theory]], whose theory is now called [[Galois theory]].

The Theories of Substitutions and Groups are among the most important

in the whole mathematical field, the study of groups and the search

for invariants now occupying the attention of many

mathematicians. The first recognition of the importance of the

combinatory analysis occurs in the problem of forming an

<math>m</math>th-degree equation having for roots $m$ of the roots of a given

<math>n</math>th-degree equation (<math>m < n</math>). For simple cases the problem goes back to [[Hudde]] (1659). [[Saunderson]] (1740) noted that the determination

of the quadratic factors of a biquadratic expression necessarily

leads to a sextic equation, and [[Le S&oelig;ur]] (1748) and [[Waring]] (1762

to 1782) still further elaborated the idea.

Lagrange first undertook a scientific treatment of the theory of

substitutions. Prior to his time the various methods of solving

lower equations had existed rather as isolated artifices than as

unified theory.

Through the great power of analysis possessed by [[Lagrange]] (1770,

1771) a common foundation was discovered, and on this was built the

theory of substitutions. He undertook to examine the methods then

known, and to show a priori why these succeeded below the quintic,

but otherwise failed. In his investigation he discovered the

important fact that the roots of all resolvents (résolvantes,

réduites) which he examined are rational functions of the roots

of the respective equations. To study the properties of these

functions he invented a "Calcul des Combinaisons." the first

important step towards a theory of substitutions. Mention should

also be made of the contemporary labors of [[Vandermonde]] (1770) as

foreshadowing the coming theory.

The next great step was taken by [[Ruffini]] (1799). Beginning like

Lagrange with a discussion of the methods of solving lower

equations, he attempted the proof of the impossibility of solving

the quintic and higher equations. While the attempt failed, it is

noteworthy in that it opens with the classification of the various

"permutations" of the coefficients, using the word to mean what

Cauchy calls a "système des substitutions conjuguées", or

simply a "système conjugué," and Galois calls a "group of

substitutions." Ruffini distinguishes what are now called

intransitive, transitive and imprimitive, and transitive and

primitive groups, and (1801) freely uses the group of an equation

under the name "l'assieme della permutazioni." He also publishes a

letter from Abbati to himself, in which the group idea is prominent.

To [[Galois]], however, the honor of establishing the theory of groups

is generally awarded. He found that if <math>r_1, r_2, \ldots r_n</math> are

the <math>n</math> roots of an equation, there is always a group of

permutations of the <math>r</math>'s such that (1) every function of the roots

invariable by the substitutions of the group is rationally known,

and (2), reciprocally, every rationally determinable function of the

roots is invariable by the substitutions of the group. Galois also

contributed to the theory of modular equations and to that of

elliptic functions. His first publication on the group theory was

made at the age of eighteen (1829), but his contributions attracted

little attention until the publication of his collected papers in

1846 (Liouville, Vol. XI). Galois is also honored as the first mathematician linking group theory with [[field theory]], whose theory is now called [[Galois theory]].

[[Cayley]] and [[Cauchy]] were among the first to appreciate the importance

of the theory, and to the latter especially are due a number of

important theorems. The popularizing of the subject is largely due

to [[Serret]], who has devoted section IV of his algebra to the theory;

to [[Camille Jordan]], whose Traité des Substitutions is a classic;

and to [[Netto]] (1882), whose work has been translated into English by

Cole (1892). [[Bertrand]], [[Hermite]], [[Frobenius]], [[Kronecker]], and [[Mathieu]]

have added to the theory. The general problem to determine the

number of groups of <math>n</math> given letters still awaits solution.

But overshadowing all others in recent years in carrying on the

labors of Galois and his followers in the study of discontinuous

groups stand [[Klein]], [[Lie]], [[Poincaré]], and [[Picard]]. Besides these

discontinuous groups there are other classes, one of which, that of

finite continuous groups, is especially important in the theory of

differential equations. It is this class which Lie (from 1884) has

studied, creating the most important of the recent departments of

mathematics, the theory of transformation groups. Of value, too,

have been the labors of Killing on the structure of groups, Study's

application of the group theory to complex numbers, and the work of

Schur and Maurer.

It was Walter Van Dyck who in 1882 gave the modern definition of a group.