Thread: Prove that if b is the only element of order 2 in G, then b E Z(G)

the center Z(G) is defined as Z(G) = {a E G | ax = xa for every x E G}
Prove that if b is the only element of order 2 in G, then b E Z(G)

Originally Posted by rainyice

the center Z(G) is defined as Z(G) = {a E G | ax = xa for every x E G}
Prove that if b is the only element of order 2 in G, then b E Z(G)

This is actually quite a neat question. What you need to do it to prove that satisfies this condition. I will just give you a hint though. Look in this thread of yours. You will use one of these three facts, along with the fact that if then . You will also have to fiddle around with the condition.

Then G has only one subgroup of order 2, then is normal in G since it doesn't have any conjugates. So then if , then since it must be in but it's not . QED