S2 in S3

We consider the subgroup $$H$$ in the group $$G$$ defined as follows.

$$G$$ is the symmetric group of degree three, which, for concreteness, we take as the symmetric group on the set $$\{ 1,2,3 \}$$.



$$H$$ is the subgroup of $$G$$ comprising those permutations that fix $$\{ 3 \}$$. In particular, $$H$$ is the symmetric group on $$\{ 1, 2\}$$, embedded naturally in $$G$$. It is isomorphic to cyclic group:Z2. As a set, $$H$$ contains two elements: $$$$ and $$(1,2)$$.

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

$$H = H_3 = \{, (1,2) \}, \qquad H_1 = \{ , (2,3) \}, \qquad H_2 = \{ , (1,3) \}$$

With this notation, $$H_i$$ is the stabilizer of $$\{ i \}$$ in the symmetric group on $$\{ 1,2,3 \}$$.

All these subgroups are $$2$$-Sylow subgroups in $$G$$.

See also subgroup structure of symmetric group:S3.

Cosets
There is a total of nine subsets of size two that arise as cosets of $$H$$ and its conjugates. Each subset, along with which subgroup it is a left or right coset of, is detailed below. Note that we use the convention that functions act on the left. The roles of left and right may get interchanged in the opposite convention.

The cosets are parametrized by ordered pairs $$(i,j) \in \{ 1,2,3\} \times \{ 1,2,3\}$$. 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 $$H_j$$.

Here is an alternative description, where the subset in a given row and given column is a left coset of its row label and a right coset of its column label. Note that this is the set of elements sending the row subgroup's fixed point to the column subgroup's fixed point:

Effect of subgroup operators
In the table below, we provide values specific to $$H$$.

Intermediate subgroups
The subgroup has prime index, hence is maximal, so there are no strictly intermediate subgroups between the subgroup and the whole group.

Smaller subgroups
The subgroup is a group of prime order, so there are no proper nontrivial smaller subgroups contianed in it.

Images under quotient maps
Under any quotient map with a nontrivial kernel, the image of the subgroup is the same as that of the whole group. This is because if the kernel is nontrivial, it must contain the cyclic subgroup $$\{, (1,2,3), (1,3,2) \}$$ of order three, and $$S_2$$ intersects each coset of this subgroup.

Automorphisms and endomorphisms: properties satisfied
For ease of reference, we take here the subgroup $$H = \{, (1,2) \}$$, though the conclusions apply for the other two conjugates as well.

Induced representations from subgroup to whole group
We consider induced representations from $$H$$ to $$G$$.

Relationship between irreducibles and those of subgroups: Frobenius reciprocity
Here, the number in a cell is the multiplicity of the column representation in the restriction of the row representation to the subgroup; equivalently, it is the multiplicity of the row representation in the induced representation from the subgroup to the whole group. These numbers are equal by Frobenius reciprocity.

Finding these subgroups inside a black-box symmetric group of degree three
We can find these subgroups in many different ways. Here is one method for a black-box group $$G$$, using SylowSubgroup to find one subgroup and then using GAP:ConjugacyClassSubgroups to find the rest:

gap> H := SylowSubgroup(G,2);; gap> L := AsList(ConjugacyClassSubgroups(G,H));; gap> H := L[1];;H1 := L[2];;H2 := L[3];;

Constructing the symmetric group and the three subgroups
Because of GAP's native implementation of symmetric groups, this is particularly easy and can be achieved using the SymmetricGroup function:

gap> G := SymmetricGroup(3);; gap> H := SymmetricGroup(2);; gap> H1 := SymmetricGroup([2,3]);; gap> H2 := SymmetricGroup([1,3]);;