Show that $A_4$ is a subgroup of $A_5.$ Let $A_n$ denote the alternating group of even permutations of $\{1,2,\ldots,n\}$.
Is $A_4$  a subgroup of $A_5$?

There are two things you want when checking if a subset is a subgroup: closure and inverses. 
(More formally, $H < G$ if for all $h_1, h_2 \in H$ we have $h_1h_2 \in H$, and for all $h \in H$, we have $h^{-1} \in H$. Some people like to condense this into one criterion and say $H$ is a subgroup if $h_1h_2^{-1} \in H$ but I find this less useful in practice.)
$A_4$ is certainly a subset of $A_5$. We already know $A_4$ is closed under the operation that $A_5$ has because that's how the group is defined. We also already know $A_4$ contains all the inverses of its elements. So yes, it's a subgroup.
