Classification of finite simple groups: Difference between revisions

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{{Group theory sidebar|Finite}}

In [[mathematics]], the '''classification of finite simple groups''' (popularly called the '''enormous theorem<ref>{{Cite web |last=published |firstauthor1=Rose Eveleth |date=2011-12-09 |title=The Funniest Theories in Physics |url=https://www.livescience.com/33628-funny-physics-theorems-names.html |access-date=2024-11-16 |website=livescience.com |language=en}}</ref>'''<ref>{{Cite web |date=2015-07-01 |title=Researchers Race to Rescue the Enormous Theorem before Its Giant Proof Vanishes |url=https://www.scientificamerican.com/article/researchers-race-to-rescue-the-enormous-theorem-before-its-giant-proof-vanishes/ |access-date=2024-11-16 |website=Scientific American |language=en}}</ref>) is a result of [[group theory]] stating that every [[List of finite simple groups|finite simple group]] is either [[cyclic group|cyclic]], or [[alternating groups|alternating]], or belongs to a broad infinite class called the [[groups of Lie type]], or else it is one of twenty-six exceptions, called [[sporadic groups|sporadic]] (the [[Tits group]] is sometimes regarded as a sporadic group because it is not strictly a [[group of Lie type]],<ref name=Conway>{{harvtxt|Conway|Curtis|Norton|Parker|Wilson|1985|loc=p. viii}}</ref> in which case there would be 27 sporadic groups). The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.

Simple groups can be seen as the basic building blocks of all [[finite group]]s, reminiscent of the way the [[prime number]]s are the basic building blocks of the [[natural number]]s (the natural number/s 0 and/or 1 cannot be built from the primes). The [[Jordan–Hölder theorem]] is a more precise way of stating this fact about finite groups. However, a significant difference from [[integer factorization]] is that such "building blocks" do not necessarily determine a unique group, since there might be many non-[[isomorphic]] groups with the same [[composition series]] or, put in another way, the [[group extension#Extension problem|extension problem]] does not have a unique solution.