To show that $A_4$ is solvable I need to show that $A_4$ is solvable.
From what i know the definition of solvable expects to give some chain of subgroups such that each subgroup in the chain is normal to the one in which it is contained and also the quotient group is abelian. 
So I tried constructing such a chain but I was not able to 

$A_4$ has a normal abelian subgroup $V=\{ id, (12)(23), (13)(24), (14)(23) \}.$ Then we have a normal series $A_4 \vartriangleright V \vartriangleright E$. Here $|V|=4$. Hence, $|A_4/V|=3$ and thus $A_4/V$ is abelian.
