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

**didact273** · February 22nd 2009, 07:54 PM

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

**TheEmptySet** · February 22nd 2009, 08:32 PM

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

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

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

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