4. (10 points) If H is a subgroup of G, then by the centralizer Z(H) of H we mean the set \{x \in G|xh = hx \text{ for all } h \in H\}.\newline(a) (5 points) Prove that Z(H) is a subgroup of G.\newline(b) (5 points) Prove that if b is the only element of order 2 in G, then $b \in Z(G)$\newlineSolution:
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Step 1: To prove that ZH is a subgroup of G, we need to show that it is closed under the group operation, contains the identity element, and contains the inverse of each of its elements. Show more…
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