A4 in S4

This article describes the subgroup $$H$$ in the group $$G$$, where $$G$$ is the symmetric group of degree four, acting on the set $$\{ 1,2,3,4 \}$$ (for concreteness) and $$H$$ is the alternating group of degree four, i.e., the subset of $$G$$ comprising the even permutations.

$$H$$ is a subgroup of index two, and its unique other coset (which is both a left coset and right coset) is the set of odd permutations.

Explicitly:

$$\! H = \{, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3), (1,2,3), (1,3,2), (1,3,4), (1,4,3), (1,2,4), (1,4,2), (2,3,4), (2,4,3) \}$$

See also element structure of symmetric group:S4 (to understand more about the elements, and which of them are even and which are odd permutations) and subgroup structure of symmetric group:S4.

Cosets
$$H$$ is a subgroup of index two, hence a normal subgroup. It has exactly two cosets: the subgroup itself and the rest of the group. Each of these is both a left coset and a right coset. The subgroup is the set of even permutations, and the other coset is the set of odd permutations. Explicitly:

$$\! H = \{, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3), (1,2,3), (1,3,2), (1,3,4), (1,4,3), (1,2,4), (1,4,2), (2,3,4), (2,4,3) \}$$

$$\! G \setminus H = \{ (1,2), (1,3), (1,4), (2,3), (2,4), (3,4), (1,2,3,4), (1,4,3,2), (1,2,4,3), (1,3,4,2), (1,4,2,3), (1,3,2,4) \}$$

For more on the element structure and interaction with conjugacy class structure, see element structure of symmetric group:S4.

Complements
There are six different candidates for a permutable complement to $$H$$ in $$G$$. Since $$H$$ is a normal subgroup of $$G$$, these are also precisely the lattice complements of $$H$$ in $$G$$. Each of these is isomorphic to the quotient group cyclic group:Z2 and in fact, in this case, they are all conjugate subgroups in $$G$$:

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

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

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 four is given, and we need to locate in that the alternating group of degree four. 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 := DerivedSubgroup(G);

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(4);; gap> H := AlternatingGroup(4);;

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