A3 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 the identity element and the two 3-cycles. It is thus the subgroup of all even permutations, i.e., the alternating group $$A_3$$. Explicitly:

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

$$H$$ is a satisfies property::normal subgroup and in fact a satisfies property::characteristic subgroup of $$G$$. It is the unique $$3$$-Sylow subgroup of $$G$$.

See also subgroup structure of symmetric group:S3.

Cosets
The subgroup is a subgroup of index two, and hence has two cosets -- the subgroup itself and the complement of the subgroup in the group:

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

In particular, the subgroup is a normal subgroup -- every left coset is a right coset and vice versa. (see also index two implies normal).

Complements
$$H$$ is a satisfies property::complemented normal subgroup of $$G$$. There are three possibilities for its permutable complement, all of which are conjugate subgroups. This can also be seen by the Schur-Zassenhaus theorem since $$H$$ is a satisfies property::normal Sylow subgroup (i.e., normal and a satisfies property::Sylow subgroup) and hence a satisfies property::normal Hall subgroup (i.e., normal and a satisfies property::Hall subgroup).

The three complements are:

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

For information on these as subgroups, see S2 in S3.

Subgroup-defining functions
The subgroup is a characteristic subgroup of the whole group and arises as a result of many common subgroup-defining functions on the whole group. Some of these are given below.

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 contained in it.

Images under quotient maps
Under any quotient map with a nontrivial kernel, the image of the subgroup is trivial. This is because the group is a monolithic group and the subgroup is the unique minimal normal subgroup in it.

Finding this subgroup inside the group as a black box
Here, a group $$G$$ that we know to be isomorphic to the symmetric group of degree three is given, and we need to locate in that the alternating group of degree three. Different ways of constructing/locating this subgroup are given below.

To assign $$H$$ to any of these, do H := followed by that. For instance:

H := SylowSubgroup(G,3);

Constructing the group and the subgroup
Because of GAP's native implementation of symmetric groups, this can be easily achieved using SymmetricGroup and AlternatingGroup:

gap> G := SymmetricGroup(3);; gap> H := AlternatingGroup(3);;

Note that double semicolons have been used to suppress confirmatory output, but you may prefer to use single semicolons.