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Asked Sep 22, 2019
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Suppose that a E G. For each y in the conjugacy class of a, let G(a y)= {g E G: gag= y}
be the set of group elements which conjugate a into y. Prove that G(a y) is a coset of CG(a)
1
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7 Suppose that a E G. For each y in the conjugacy class of a, let G(a y)= {g E G: gag= y} be the set of group elements which conjugate a into y. Prove that G(a y) is a coset of CG(a) 1

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Expert Answer

Step 1

To prove that the given subset G(a -> y) is a coset of the centralizer of a in G.

Step 2

Recall the general fact about subgroups and cosets in any group G

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H a subgroup of G Fact SG is a cos et of H g,heSgheH

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Step 3

Apply this criterion to subset S and ...

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Let S = G(a y) {g eG: gag = y} and H =C(a) Centralicer of a in G =r G: axa

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