# Is the neutral element unique in non-abelian groups?

I found different definitions of a 'group'. One stated that the neutral element has to be unique, the other only that it had to exist. I assumed that this means that the neutral element will be unique if it exists, but I was only able to prove this for commutative groups.

Could there possibly be multiple neutral elements in non-abelian groups? If not, how can I prove it?

• Oops, I forgot that the neutral element can be both on the right or on the left - now it's easy! :)
– mafu
Apr 9, 2014 at 21:58

A neutral element $e$ will be unique if it is both a "left-neutral" element and a "right-neutral" element, i.e. $eg = ge = g$ for all $g \in G$. I think this is the usual definition of a neutral element. If you only require "left-neutrality" as I assume you were working with, it is probably not true.
Suppose $e$ and $e'$ are two neutral elements. Then $ee'=e'$ because $e$ is neutral and $ee'=e$ because $e'$ is neutral. So $e=e'$.
Given two identity elements $e$ and $e'$, we have $e=ee'=e'$. Don't need commutativity here.