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3. question let g be a group and let ...

# Question: question let g be a group and let ...

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Question:

Let G be a group and let ◇ ∶ G × G → G, (a, b) → aba-1 be the action of G on G by conjugation.

Define Z(G) ∶= {a ∈ G ∣ ab = ba for all b ∈ G}.

(a) Show that Z(G) = Fix◇ G(G).

(b) Suppose G is a p-group with ∣G∣ > 1. Show that ∣Z(G)∣ > 1. (Hint: Use (a) and the Fixed Point Formula)