Prove that, if G is a group and each element of G has order at most 2, then G is abelian.

Thanks.

- Mar 1st 2011, 02:37 AMfeyomiProve that G is abelian given that each element of G has order at most 2.
Prove that, if G is a group and each element of G has order at most 2, then G is abelian.

Thanks. - Mar 1st 2011, 03:16 AMDrSteve
Since $\displaystyle a^2=e$, $\displaystyle a=a^{-1}$ for all $\displaystyle a$. Now look at the equation $\displaystyle (ab)^2=e$

- Mar 2nd 2011, 07:14 PMJJMC89
$\displaystyle e$ 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.