Prove that if $\displaystyle G$ is a group, then the only element $\displaystyle g \in G$ such that $\displaystyle g^2=g$ is 1.
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Originally Posted by didact273 Prove that if $\displaystyle G$ is a group, then the only element $\displaystyle g \in G$ such that $\displaystyle g^2=g$ is 1. Suppose that there is another element such that $\displaystyle a^2=a$ Since G is a group a must have an inverse so multiply both sides by a inverse $\displaystyle a^{-1}(a^2)=a^{-1}a \iff a=1$
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