Thread: Quick question about identity element

Say I have a group and an unknown binary operation , and . ( being the identity element)

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

Originally Posted by tangibleLime

Say I have a group and an unknown binary operation , and . ( being the identity element). Is it true that the only value can be is the identity element, ?

Not true. Choose for example , we have .

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

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").

What is true, however, is that if then is the identity. This is true because,