= Solution
The <alternating group> $A_4$ contains
$$
V=\{1,(12)(34),(13)(24),(14)(23)\}.
$$
The product of any two different nonidentity elements is the third, and each squares to the identity, so $V$ is a <Klein four-group>. Conjugation by a permutation replaces the labels in each cycle, preserving the property of being a product of two disjoint transpositions. Thus $V$ is a <normal subgroup> of $A_4$. Its order is $4$, strictly between $1$ and $|A_4|=12$. \b[$A_4$ is not simple.]
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