Is H2 the same as the 4th alternating group 
Would H2=A4 as if we apply sigma^5 to both sides we get sigma^12=1 which is true for all elements of the 4th alternating group by Lagrange's theorem.

$H_2\subset A_4$, by definition.   Conversely,  let $\sigma \in A_4$.  Then $\sigma ^{12}=1$ (since $\mid A_4\mid=12$) $\implies \sigma ^7=\sigma ^{-5}\implies \sigma \in H_2$. Thus $A_4\subset H_2$. Thus $H_2=A_4$
