Question
Prove that $a=a^2$ in a group if and only if $a=e$
Attempt Suppose that $a^2=a$ such that $a\in G$ , that is $(G, *)$ Then $a*e=e*a=a$ Implies that $ae=ea=a$ Now suppose that $a≠e$ Since $a^{-1}a=aa^{-1}=e$ Then $a(aa^{-1})=e$ $aa=e$ $a^2=e$ Now suppose that $a=e$ Then $a=a^2$
Please help me with your solution so that I would know that what I have done is correct.
In every group $G$ we have the cancellation law. So we can "cancel" an $a$, by multiplying with $a^{-1}$ on both sides of the equation $a=a^2=aa$, to obtain that $a^{-1}a=a^{-1}aa$, which says that $e=a$.