# S3 in S4

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) symmetric group:S3 and the group is (up to isomorphism) symmetric group:S4 (see subgroup structure of symmetric group:S4).
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We consider the subgroup $H$ in the group $G$ defined as follows. $G$ is the symmetric group of degree four, which, for concreteness, we take as the symmetric group on the set $\{ 1,2,3,4 \}$. $H$ is the subgroup of $G$ comprising those permutations that fix $\{ 4 \}$. In particular, $H$ is the symmetric group on $\{ 1, 2,3 \}$, embedded naturally in $G$. It is isomorphic to symmetric group:S3. $H$ has order $6$.

There are three other conjugate subgroups to $H$ in $G$ (so the total conjugacy class size of subgroups is 4). The other subgroups are the subgroups fixing $\{ 1 \}$, $\{ 2 \}$, and $\{ 3 \}$ respectively.

The four conjugates are: $\! H = H_4 = \{ (), (1,2), (1,3), (2,3), (1,2,3), (1,3,2) \}$ $\! H_1 = \{ (), (2,3), (3,4), (2,4), (2,3,4), (2,4,3) \}$ $\! H_2 = \{ (), (1,3), (3,4), (1,4), (1,3,4), (1,4,3) \}$ $\! H_3 = \{ (), (1,2), (2,4), (1,4), (1,2,4), (1,4,2) \}$

## Cosets

There are four left cosets and four right cosets of each subgroup. Each left coset of a subgroup is a right coset of one of its conjugate subgroups. This gives a total of 16 subsets.

The cosets are parametrized by ordered pairs $(i,j) \in \{ 1,2,3,4 \} \times \{ 1,2,3,4 \}$. The coset parametrized by $(i,j)$ is the set of all elements that send $i$ to $j$. This is a left coset of $H_i$ and a right coset of $H_j$.

## Complements

There is a unique normal complement that is common to all the subgroups. This is the subgroup normal Klein four-subgroup of symmetric group:S4: $\! K := \{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$

There is also a conjugacy class of subgroups each of which is a permutable complement to each of the $H_i$s. These are cyclic four-subgroups of symmetric group:S4, and these are: $\{ (), (1,2,3,4), (1,3)(2,4), (1,4,3,2) \}, \qquad \{ (), (1,3,2,4), (1,2)(3,4), (1,4,2,3) \}, \qquad \{ (), (1,2,4,3), (1,4)(2,3), (1,3,4,2) \}$

Note that the fact that these are permutable complements can be understood as a special case of Cayley's theorem. See also every group of given order is a permutable complement for symmetric groups, which says that any finite group of order $n$ is, via the Cayley embedding, a permutable complement to $S_{n-1}$ in $S_n$.

Apart from these, each of the $H_i$s has a number of lattice complements:

• Any subgroup generated by double transposition in S4 is a lattice complement to each $H_i$ in the whole group. Thus, each $H_i$ has three such lattice complements.
• For each $H_i$, a subgroup of order three not contained in that $H_i$ is a lattice complement to it. Thus, each $H_i$ has three such lattice complements, because one of the four subgroups of order three is contained in that $H_i$.

### Properties related to complementation

Property Meaning Satisfied? Explanation Comment
retract has a normal complement Yes subgroup $K$ above is a normal complement
permutably complemented subgroup has a permutable complement Yes normal complement is permutable complement too
lattice-complemented subgroup has a lattice complement Yes normal complement is lattice complement too
complemented normal subgroup normal subgroup with permutable complement No not normal itself

## Arithmetic functions

Function Value Explanation
order of whole group 24
order of subgroup 6
index of subgroup 4
size of conjugacy class of subgroup (=index of normalizer) 4 see above for list of conjugates
number of conjugacy classes in automorphism class of subgroup 1 the whole group is a complete group, so the conjugation actions are precisely the automorphisms.
size of automorphism class of subgroup 4 same as size of conjugacy class