Prove that, if G is a group and each element of G has order at most 2, then G is abelian.
Thanks.
$\displaystyle e$ is the only element of order $\displaystyle 1$.
if $\displaystyle g,h \in G$ then $\displaystyle gh \in G$
so $\displaystyle (gh)^2 = e$
and $\displaystyle (gh)^2 = ghgh$
so $\displaystyle ghgh = e$
$\displaystyle ghghh = h$
$\displaystyle ghg = h$
$\displaystyle ghgg = hg$
$\displaystyle gh = hg$
Therefore $\displaystyle G$ such that all elements have order of at most $\displaystyle 2$ is Abelian.