# Thread: group and identity help

1. ## group and identity help

Suppose that (G, *) is a group, with identity e, where cardinality of G is even. Show that there must be an element g in G, with g not equal to e and g*g=e.

2. Originally Posted by antoniowilliams46
Suppose that (G, *) is a group, with identity e, where cardinality of G is even. Show that there must be an element g in G, with g not equal to e and g*g=e.
If $\displaystyle g\ne g^{-1}$ then $\displaystyle (g^{-1})\ne (g^{-1})^{-1}$. In other words the elements which aren't idempotent come in pairs.