Classification of finite simple groups

Statement

Classification of finite simple groups. Every finite simple group is isomorphic to a cyclic group $C_p$ of prime order; an alternating group $A_n$ with $n\geq5$ ; a simple group in the Lie-type classification, including the Tits group ${}^2F_4(2)'$ ; or one of the 26 sporadic simple groups.

Reading the families

The finite-group table separates the two elementary families, six classical Lie-type families, ten exceptional Lie-type families, the Tits boundary case, and the 26 sporadics. Field sizes, ranks, central quotients, and small nonsimple exceptions are part of each family's constraints. The split construction and twisted construction explain the Lie-type entries.

Some small groups have more than one family presentation. A table tile therefore names a family or a specified construction, rather than asserting that all tiles are pairwise nonisomorphic.

What the theorem does not classify

The Jordan–Hölder theorem makes the multiset of simple composition factors of a finite group well-defined. Those factors do not determine the original group: $C_4$ and $C_2\times C_2$ each have two factors isomorphic to $C_2$ , but only $C_4$ has an element of order four.

References

P. J. Cameron, Group Theory revision notes, §5.4, p. 26: the classification statement; §§5.1–5.2, pp. 22–24: composition factors.
R. Solomon, The Classification of the Finite Simple Groups: A Progress Report, Notices of the AMS 65 (2018), pp. 646–651, especially the classification theorem on p. 648.