Let $G$ be a group of order $8$ and $x∈G$ such that order of $x$ is $4$. I want to show that $x^2∈Z(G)$.
Can anyone help?
Let $G$ be a group of order $8$ and $x∈G$ such that order of $x$ is $4$. I want to show that $x^2∈Z(G)$.
Can anyone help?
Here's a more elementary proof than the one I gave in my comment to the question. (I give a series of hints, since I suspect this is a homework question.)
Define $X$ to be the subgroup generated by $x$.
First, show that for any $g\in G$, $g^2\in X$. This shows that $X$ is a normal subgroup of $G$; in particular any conjugate of $x$ is either $x$ or $x^{-1}$. (There are three statements that need proof.)
Now, compute $g^{-1}x^2g$ and rearrange terms to prove that $x^2$ commutes with every element of $G$.