Let $G$ be the Klein 4-group. What four permutations in $S_4$ form a subgroup $\cong G$? I know that $G$ is an abelian group and is composed of $\{e, a, b, ab\}$. I also know that every non-identity element has order 2. But I'm not sure how to write out the permutations $F_e, F_a, F_{ab}$ I do know $F_b$ would be $\{b, ab, e, a\}$ which is $(13)(24)$, right?
 A: You can embed the Klein 4-group (which is not cyclic, since it contains no element of order $4$!) into $S_4$ in four ways: 
$$\langle (12),(34)\rangle=\{1,(12),(34),(12)(34)\} $$
$$\langle (13),(24)\rangle=\{1,(13),(24),(13)(24)\} $$
$$\langle (14),(23)\rangle=\{1,(14),(23),(14)(23)\} $$
$$\langle(12)(34),(14)(23)\rangle=\{1,(12)(34),(14)(23),(13)(24)\}$$
but only the last subgroup happens to be normal.
A: One way to solve this problem is to use the left regular representation.  $V$ acts on itself by left multiplication.  First, we choose a numbering of the elements of $V$.  Somewhat arbitrarily, I choose to label $1,a,b,ab$ by $1,2,3,4$, respectively.  First note that $1$ must act as the identity permutation.  We observe that $a$ acts by left multiplication by sending
\begin{align*}
a: 1 &\mapsto a\\
a &\mapsto a^2 = 1\\
b &\mapsto ab\\
ab &\mapsto b \, .
\end{align*}
Recalling our numbering we see that the permutation $\sigma_a$ corresponding to $a$ sends
\begin{align*}
\sigma_a: 1 &\mapsto 2\\
2 &\mapsto 1\\
3 &\mapsto 4\\
4 &\mapsto 3
\end{align*}
so $\sigma_a = (1\ 2)(3\ 4)$.  Proceeding similarly with $b$ and $ab$, we find $\sigma_b = (1\ 3)(2\ 4)$ and $\sigma_{ab} = (1\ 4)(2\ 3)$.
Note that choosing a different numbering for the elements of $V$ will yield different permutations.  In fact changing the numbering conjugates all the permutations.
A: An easy way to do this is just to examine the permutations of order 2, since all elements of the Klein-4 group have order 2. These are just the set of all transpositions and disjoint transposition.
Note that the transpositions alone do not form a normal subgroup, so we have all of the pairs of disjoint transpositions: 
$$\{(),(12)(34),(13)(24),(14)(23)\}.$$ 
