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Prove that if $g^2=e$ for all g in G then G is Abelian.
This is how I proved it:
Abelian means that the following axioms hold: Associativity, Existence of Identity and inverse elements, commutativity.
1) Associativity:
For some element $h \in G$, we have (hg)g = h(gg) = h. Therefore holds
2) Existence of identity
From definition: $g^2 = I_G$
3) Existence of inverse
As G is already a group, thus there exists a $g^{-1}$ such that $g^{-1} \cdot g = 1_G$
4) Commutativity
$hg^2 = g^2h = I_Gh = h$
Thus commutativity holds.
Is this proof correct?