Find an element $a$ from some group such that $|a|=6$ and $C(a) \neq C(a^3)$ Find an element $a$ from some group $G$ such that $|a|=6$ and $C(a) \neq C(a^3)$.
$|a|$ is the order of $a$.
$C(a)$ is the Centraliser of $a$ in $G$.
This question is from Contemprary Abstract Algebra by Joseph A Gallian 4th edition,
Chapter 3, Question 24.

We want $a$ to have order $6$? Let's try $a=(1\ 2\ 3\ 4\ 5\ 6)$ in the group $S_6$. Let's see, $a^3=(1\ 4)(2\ 5)(3\ 6)$ right? Does $C(a)$ mean the set of group elements that commute with $a$? So we're looking for something that commutes with $a^3$ but not with $a$? Hmm. The transposition $(1\ 4)$ commutes with $a^3$, doesn't it? I wonder if it commutes with $a$.
