A monoid where square of all elements are 1 is abelian

Question

The following problem gives me a very hard time:

Let $M$ be a monoid with $a^2 = 1$ for $a \in M$. Show that $M$ is abelian.

It looks so simple as a monoid only needs to be associative and must have a neutral element (here $1$). So there are not much things to try. However, after some hours of trying I have to admit that I don't know what to try next.

Maybe somebody of you can give me a hint.

Kind regards!

Your title is misleading. An element $a$ is idempotent if $a^2 = a$, not if $a^2 = 1$. — J.-E. Pin, Mar 22 '17 at 08:13
Calling it a monoid is a red herring: obviously it is a group, too. — rschwieb, Mar 22 '17 at 17:25

score 3 · Accepted Answer · answered Mar 21 '17 at 21:09

3

$1=abab\to b=ababb \to b=aba \to ab =aaba \to ab =ba $

answered Mar 21 '17 at 21:09

A monoid where square of all elements are 1 is abelian

1 Answers1