I'm trying to prove that the subgroup $\{I , (12)(34), (13)(24), (14)(23)\} = Z_2 ⊗ Z_2$ is an invariant (normal) subgroup of $A_4$, where $(12)(34)$ and so on is a notation to indicate permutations (for example $(12)(34)$ would be the permutation $(2\quad 1\quad 4\quad 3)$), I used the equality sign to indicate that the two groups are isomorphic, and $A_4$ is the group of permutations of 4 objects.
The book says to "verify" this, and maybe the author means actually trying all 24 possible permutations ($p\in A_4$) and performing the operation $p^{-1}gp$ for $g\in Z_2 ⊗ Z_2$, but I was trying to come up with some argument in order not to do it manually and I couldn't think of anything.
Can someone help?
