Group commute properties

Question

Let $G$ be a group

If $y\in G$ commutes with some $x\in G$ then $y$ commutes with powers of $x$ i.e $yx^n =x^ny$.

I know this is true but is the other way around true?

If $y\in G$ commutes with a power of some $x\in G$ say $x^n$ then $y$ commutes with $x$ i.e $yx=xy$

This is not true. Consider the dihedral group for example. – Kal S. May 21 '14 at 00:48 — Kal S., May 21 '14 at 00:48

score 4 · Accepted Answer · answered May 21 '14 at 00:47

4

Let $n$ be the order of $x$. You know $yx^n=ye=y=ey=x^ny$, but in general it's not true that $xy=xy$.

answered May 21 '14 at 00:47

score 3 · Answer 2 · answered May 21 '14 at 00:48

Nope: Every non-abelian finite group produces counterexamples to this; if $|G| = n$, then the fact that $x^n = e$ for all $y \in G$ implies that

$$yx^n = x^n y$$

In fact, this gives counterexamples in every group where there is a non-central element of finite order.

2 Answers2