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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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V.7 The Classification of Finite Simple Groups

Martin W. Liebeck

A finite group G is said to be simple if its only normal subgroups are the identity subgroup and G itself. To some extent, simple groups play an analogous role in finite group theory to that of prime numbers in number theory: just as the only factors of a prime p are 1 and p itself, so the only factor groups of a simple group G are the identity group 1 and G itself. The analogy runs a bit deeper: just as every positive integer (greater than 1) is a product of a collection of primes, so every finite group is “built” from a collection of simple groups, in the following sense. Let H be a finite group, and choose a maximal normal subgroup H1 of H (this means that H1 is not the whole ...

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