Show that $\{ I,(12)(34),(13)(24),(14)(23)\}$ subgroup of $A_4$ is invariant

Question

Let $H:= (h_1,h_2,...)$ be a subgroup of $G$. Take any element g, not in $H$. Then the set of elements $(g^{-1}h_1 g,g^{-1}h_2 g,...)$ forms a subgroup, which we denote by $g^{-1}Hg$. If H and $g^{-1}Hg$ are the same for all $g \in G$, then $H$ is invariant.

How can I show that a subgroup of $A_4$,$$\{ I,(12)(34),(13)(24),(14)(23)\}$$ forms an invariant group and $$\{I,(12)(34),(13)(24),(14)(23) \}\cong\Bbb Z_2 \otimes\Bbb Z_2 $$

Reference: Group Theory in a Nutshell for Physicists, A.Zee

I tried several other $A_4$ elements such as $(123), (142),$ but I somehow cannot write $(142)^{-1}(12)(34)(142)$ into another element of the subgroup above...

Thanks in Advance!

Answer 1

Since the conjugation $\beta^{-1}\alpha\beta$ of a permutation $\alpha$ by another permutation $\beta$ preserves the cycle structure of $\alpha$, and the candidate subgroup $H$ in question contains all elements of the form $(ab)(cd)$ (together with the identity), we must have that $H\lhd A_4$, which is equivalent to $H$ being invariant.

Since $H$ is closed under multiplication, $H< A_4$. Thus $H$ is, in particular, a group. Each of its nontrivial elements has order two; therefore, by the classification of groups of order four, $H\cong \Bbb Z_2^2$.