1. ## Quick question about identity element

Say I have a group $G$ and an unknown binary operation $*$, and $a*a=e, \forall a \in G$. ( $e$ being the identity element)

Is it true that the only value $a$ can be is the identity element, $e$?

2. ## Re: Quick question about identity element

Originally Posted by tangibleLime
Say I have a group $G$ and an unknown binary operation $*$, and $a*a=e, \forall a \in G$. ( $e$ being the identity element). Is it true that the only value $a$ can be is the identity element, $e$?
Not true. Choose for example $(\mathbb{Z}_2,+)$ , we have $0+0=1+1=0$ .

3. ## Re: Quick question about identity element

Ah, didn't consider that. Thanks for the response!

4. ## Re: Quick question about identity element

what you can say, if a*a = e,and a is NOT the identity, is that a is of order 2, and that a is its own inverse (showing that elements of order 2 and of order 1(the identity is the only element of order 1) share this property).

a good example of an element of order 2 is the following group, which is a subset of the actions you can perform on a light switch:

{flip the switch, do nothing} and the group operation is: first do one thing, then the other.

"flip the switch" is an element of order 2: flip the switch, then flip the switch = do nothing.

(to be honest, this is in all actuality just Z2 in disguise, and is one of the basic ways in which logical circuits can "do math").

5. ## Re: Quick question about identity element

What is true, however, is that if $a\ast a=a$ then $a$ is the identity. This is true because,

$a=a^{-1}\ast a\ast a=a\ast a^{-1}=e$