# Thread: Help with proof: only element in a group with g^2=g

1. ## Help with proof: only element in a group with g^2=g

Prove that if $G$ is a group, then the only element $g \in G$ such that $g^2=g$ is 1.

2. Originally Posted by didact273
Prove that if $G$ is a group, then the only element $g \in G$ such that $g^2=g$ is 1.
Suppose that there is another element such that

$a^2=a$ Since G is a group a must have an inverse so multiply both sides by a inverse

$a^{-1}(a^2)=a^{-1}a \iff a=1$