# abelian subsets

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Let G be a group and S any subset of G. Prove that C_G (S) = {g in G such that gs = sg for all s in S} is a subgroup of G. Prove that Z (G) (center of G) = C_G (S) is abelian and is a normal subgroup of G.

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#### Solution Preview

Proof:

First, I show that C_G(S) is a subgroup of G.

We consider any g, h in C_G(S), then for any s in S, we have

gs = sg and hs = sh

Then hsh^(-1) = s and thus sh^(-1) = ...

#### Solution Summary

This solution proves that Z (G) (center of G) = C_G (S) is abelian and is a normal subgroup of G.

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