Suppose G is a group which has only one element a such that |a| = 2. Prove that xa = ax, for all x ∈ G.

Question

I know the following are true.

1) There is an inverse of a 2) There is an identity element (e*a) = a

In this case, e = 1 and the inverse of a is 1/|2|. However, if a is the only element in G and a is |2|, wouldn't that imply that x can only be x=|2|, for x ∈ G?

I'm not quite sure how to begin the proof either, I'm used to starting with "ab" cases and a binary function and then working the case down to ab=ba.

Answer 1

Hint: what is the order of conjugate of a in G?

Answer 2

Conjugate also has same order as of a. Thus $xax^{-1}=a$, for all $x$ belongs to $G$ ,this implies $xa=ax$.