If $a$ is the only element of order $2$ in a group, show that $a$ is in the center of the group.

Question

If $a$ is the only element with order $2$ in a group $G$, then $a \in Z(G)$.

I'm studying for a test and I can't figure out how to prove it. What kind of methods might I try to solve this problem? I'd prefer suggestions rather than a proof, so that I can figure it out for myself.

score 2 · Answer 1 · answered Feb 08 '16 at 19:24

2

Hint: conjugate elements have the same order.

answered Feb 08 '16 at 19:24

Nicky Hekster

49,281

Sadly, I am still lost. – Zachary F Feb 08 '16 at 20:17
1

If $|G|=2$, you're done. Otherwise, suppose $x$ is your element of order two. Then there exists some $g\in G\setminus{1}$ such that $x\neq g$. Consider $gxg^{-1}$ (the "conjugate" of $x$ by $g$). – user1729 Feb 08 '16 at 20:21
Never mind, I figured it out :D – Zachary F Feb 08 '16 at 20:24
OK Zachary, well done. I will give you another one to think about: let $G$ be a group with a unique element of order $n \gt 1$. Then $n=2$. – Nicky Hekster Feb 08 '16 at 21:28

If $a$ is the only element of order $2$ in a group, show that $a$ is in the center of the group.

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