Classification of finite simple groups

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the classification of finite simple groups (popularly called the enormous theorem) is a result of <a href="/facts/Group_theory/NfowcI5H">group theory</a> stating that every <a href="/facts/List_of_finite_simple_groups/J4YVCeWt">finite simple group</a> is either <a href="/facts/Cyclic_group/JL5zfFuL">cyclic</a>, or <a href="/facts/Alternating_groups/yethKwyo">alternating</a>, or belongs to a broad infinite class called the <a href="/facts/Groups_of_Lie_type/gmboug78">groups of Lie type</a>, or else it is one of twenty-six exceptions, called <a href="/facts/Sporadic_groups/hjNHTKVv">sporadic</a> (the <a href="/facts/Tits_group/1WImHdot">Tits group</a> is sometimes regarded as a sporadic group because it is not strictly a <a href="/facts/Group_of_Lie_type/gmboug78">group of Lie type</a>, 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 <a href="/facts/Finite_group/vv2tvogB">finite groups</a>, reminiscent of the way the <a href="/facts/Prime_number/L20FLkgn">prime numbers</a> are the basic building blocks of the <a href="/facts/Natural_number/u65og8JE">natural numbers</a>. The <a href="/facts/Jordan%25E2%2580%2593H%25C3%25B6lder_theorem/3C7CHH3k">Jordan–Hölder theorem</a> is a more precise way of stating this fact about finite groups. However, a significant difference from <a href="/facts/Integer_factorization/EjMCTz2t">integer factorization</a> is that such "building blocks" do not necessarily determine a unique group, since there might be many non-<a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> groups with the same <a href="/facts/Composition_series/3C7CHH3k">composition series</a> or, put in another way, the <a href="/facts/Group_extension/pGoe0l4d">extension problem</a> does not have a unique solution.
<a href="/facts/Daniel_Gorenstein/tBgFlDpt">Daniel Gorenstein</a> (1923–1992), <a href="/facts/Richard_Lyons_(mathematician)/HqnVz6He">Richard Lyons</a>, and <a href="/facts/Ronald_Solomon/vh2Kayk5">Ronald Solomon</a> are gradually publishing a simplified and revised version of the proof.

Classification of finite simple groups open-in-new

Classification of finite simple groups