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UNFORMATTED ATTACHMENT PREVIEW
MTH 461: Survey of Modern Algebra, Spring 2021 Lecture 18 Cayley’s Theorem How can we understand all finite groups? One way, which we return to later, is to try and classify, i.e. list, all groups up to isomorphism. Here is another way to “understand all finite groups”: show that all such groups are isomorphic to subgroups of groups we understand. This can be done using the following theorem. § Cayley’s Theorem: Let G be a finite group, and |G| “ n. Then there is a 1-1 homomorphism φ ∶ G Ñ Sn . In particular, G is isomorphic to a subgroup of Sn . Before proving this theorem, let’s see why it does what we want. Given any homomorphism φ ∶ G Ñ G1 we can define a new homomorphism ψ ∶ G Ñ impφq by setting ψpaq “ φpaq for all a P G; we have only changed the definition of the target group. The homomorphism ψ ∶ G Ñ impφq has the

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