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:

$H=H_4=\{(),(1,2),(1,3),(2,3),(1,2,3),(1,3,2)\}$

$H_1=\{(),(2,3),(3,4),(2,4),(2,3,4),(2,4,3)\}$

$H_2=\{(),(1,3),(3,4),(1,4),(1,3,4),(1,4,3)\}$

$H_3=\{(),(1,2),(2,4),(1,4),(1,2,4),(1,4,2)\}$

See also subgroup structure of symmetric group:S4.

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:

$K:=\{(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}$

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:

$\{(),(1,2,3,4),(1,3)(2,4),(1,4,3,2)\},\{(),(1,3,2,4),(1,2)(3,4),(1,4,2,3)\},\{(),(1,2,4,3),(1,4)(2,3),(1,3,4,2)\}$

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

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