Subgroups of a permutation group The permutation group $S_{4}$ is defined as the group of all possible permutations of [1234].
i) Find the number of subgroups of $S_{4}$ that have order 2.
ii) A: { [1234], [2143], [3412], [4321] } and B: { [1234], [1243], [2134], [2143] }. Which of A and B are subgroups of $S_{4}$?
Trying to teach myself a Further Maths module on Groups is proving difficult when none of my teachers know the syllabus, any help would be appreciated! Thanks.
 A: (i) The order of a permutation is easy to compute given its decomposition as a product of disjoint cycles: it is the l.c.m. of the orders of the cycles, i.e. the l.c.m. of the lengths of the cycles.
Hence a permutation of order $2$ is a product of disjoint transpositions. In $S_4$ it the product of at most $2$ disjpoint transpositions.
We have $\dbinom 42=6$ transpositions and as many products of disjoint transpositions.
(ii) If the notation is that of $4$-cycles, neither $A$ nor $B$ are subgroups.
Indeed, in $A$, let $\gamma=(1234),\enspace \gamma'=(2143)$. Then $\,(4321)=\gamma^{-1}=\gamma^3, \enspace (3412)= \gamma'^{-1}= \gamma'^3$. However $\gamma$ and $\gamma'$ have order $4$. If $A$ were a subgroup, it should contain $\gamma^2=(13)(24)$.
For $B$, let $\gamma$ be as above and $\gamma'=(1243)$. Then $B$ contains $\gamma^{-1}$ and $\gamma'^{-1}$, but doesn't contain $\gamma^2$, nor $\gamma'^2$.
A: i) You want all transpositions (ab), and all pairs of disjoint transpositions (ab)(cd).
ii) Assuming that [2143] means, in cycle notation, (12)(34), then A is a subgroup.
A: Hints:
i) If a subgroup has order two then there are exactly one trivial and one non-trivial element. Furthermore the nontrivial element $a\in G$ must fulfill $a^2=e$. So just check with of the elements of $S_4$ fulfill this requirement.
ii) Just check the subgroup axioms for $A$ and $B$.
