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 H_{1} of *H* (this means that *H*_{1} is not the whole ...

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