A3 in S3

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) cyclic group:Z3 and the group is (up to isomorphism) symmetric group:S3 (see subgroup structure of symmetric group:S3).
The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z2.
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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 normal subgroup and in fact a 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

COMPLEMENTS TO NORMAL SUBGROUP: TERMS/FACTS TO CHECK AGAINST:
TERMS: permutable complements | permutably complemented subgroup | lattice-complemented subgroup | complemented normal subgroup (normal subgroup that has permutable complement, equivalently, that has lattice complement) | retract (subgroup having a normal complement)
FACTS: complement to normal subgroup is isomorphic to quotient | complements to abelian normal subgroup are automorphic | complements to normal subgroup need not be automorphic | Schur-Zassenhaus theorem (two parts: normal Hall implies permutably complemented and Hall retract implies order-conjugate)

$H$ is a 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 normal Sylow subgroup (i.e., normal and a Sylow subgroup) and hence a normal Hall subgroup (i.e., normal and a Hall subgroup).

The three complements are:

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

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

Properties related to complementation

Property	Meaning	Satisfied?	Explanation	Comment
complemented normal subgroup	normal subgroup having a permutable complement	Yes
complemented characteristic subgroup	characteristic subgroup having a permutable complement	Yes
permutably complemented subgroup	has a permutable complement	Yes
lattice-complemented subgroup	has a lattice complement	Yes
retract	has a normal complement	No
direct factor	normal subgroup having a normal complement	No

Arithmetic functions

Function	Value	Explanation
order of whole group	6
order of subgroup	3
index	2
size of conjugacy class	1
number of conjugacy classes in automorphism class	1

Effect of subgroup operators

Function	Value as subgroup (descriptive)	Value as subgroup (link)	Value as group
normalizer	the whole group	--	symmetric group:S3
centralizer	the subgroup itself	current page	cyclic group:Z3
normal core	the subgroup itself	current page	cyclic group:Z3
normal closure	the subgroup itself	current page	cyclic group:Z3
characteristic core	the subgroup itself	current page	cyclic group:Z3
characteristic closure	the subgroup itself	current page	cyclic group:Z3
commutator with whole group	the subgroup itself	current page	cyclic group:Z3

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.

Subgroup-defining function	Meaning in general	Why it takes this value
derived subgroup	subgroup generated by commutators of all pairs of group elements, smallest subgroup with abelian quotient	The quotient is cyclic group:Z2, which is abelian; no other subgroup has abelian quotient. We can also explicitly compute all commutators -- these are precisely the identity element and the two 3-cycles.
hypocenter	subgroup at which the lower central series of the whole group stabilizes	It is the derived subgroup and its commutator with the whole group equals itself.
socle	join of all minimal normal subgroups	The subgroup is the unique minimal normal subgroup (i.e., monolith) -- the group is a monolithic group
Fitting subgroup	join of all nilpotent normal subgroups	The subgroup is the unique nontrivial abelian normal subgroup
3-Sylow core	largest normal subgroup whose order is a power of 3; normal core of any 3-Sylow subgroup	The subgroup is the unique normal 3-Sylow subgroup
3-Sylow closure	normal closure of any 3-Sylow subgroup	The subgroup is the unique normal 3-Sylow subgroup
Brauer core	largest normal subgroup of odd order	(same reason as 3-Sylow core; order has two prime factors 2 and 3)
Jacobson radical	intersection of all maximal normal subgroups	The subgroup is the unique maximal normal subgroup -- the group is a one-headed group

Description in alternative interpretations of the whole group

Interpretation of $G$	Corresponding interpretation of $H$
As the dihedral group of degree three, order six	The cyclic part comprising rotations, i.e., orientation-preserving elements.
As the general linear group of degree two over field:F2	The subgroup comprising the semisimple elements, although the ones of order three are not diagonalizable over field:F2 and can be diagonalized only over field:F4. Note that it is a feature of field:F2 that semisimple elements form a multiplicative subgroup -- they do not form a multiplicative subgroup for larger field sizes. CAUTION: The subgroup is not the special linear group of degree two! In fact, the special linear group coincides with the whole general linear group.
As the general affine group:GA(1,q) where $q = 3$	The subgroup comprising translations, i.e., where the scaling factor is $1$.

Related subgroups

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.

Subgroup properties

Invariance under automorphisms and endomorphisms: properties

Property	Meaning	Satisfied?	Explanation	Comment
normal subgroup	equals all its conjugate subgroups	Yes	Kernel of sign homomorphism; alternating group is always normal in corresponding symmetric group	Also follows from the fact that index two implies normal
characteristic subgroup	invariant under all automorphisms	Yes	normal Sylow subgroups are characteristic
coprime automorphism-invariant subgroup	invariant under all coprime automorphisms, i.e., automorphisms whose order is coprime to that of the group	Yes
cofactorial automorphism-invariant subgroup	invariant under all cofactorial automorphisms, i.e., automorphisms whose order has no prime factors other than those in the group	Yes
subgroup-cofactorial automorphism-invariant subgroup	invariant under all automorphisms whose order has no prime factors other than those in the subgroup	Yes
fully invariant subgroup	contains its image under any endomorphism of whole group	Yes	normal Sylow subgroups are fully invariant

Resemblance-based properties

Property	Meaning	Satisfied?	Explanation	Comment
Sylow subgroup		Yes
normal Sylow subgroup		Yes
Hall subgroup		Yes
normal Hall subgroup		Yes
order-conjugate subgroup	conjugate to all subgroups of the same order	Yes	follows from being a Sylow subgroup, since Sylow implies order-conjugate
order-isomorphic subgroup	isomorphic to all subgroups of the same order	Yes	(via order-conjugate, also obvious since has prime order)

Advanced properties related to invariance/resemblance

Property	Meaning	Satisfied?	Explanation	Comment
verbal subgroup		Yes
isomorph-free subgroup	no other isomorphic subgroup	Yes
order-unique subgroup	no other subgroup of that order	Yes
normal subgroup having no nontrivial homomorphism to its quotient group		Yes
homomorph-containing subgroup	contains any homomorphic image of it in the whole group	Yes	normal Sylow subgroups are homomorph-containing
variety-containing subgroup		Yes
quotient-subisomorph-containing subgroup		Yes
image-closed fully invariant subgroup		Yes	(via verbal)
intermediately fully invariant subgroup		Yes	(via variety-containing)
image-closed characteristic subgroup		Yes
intermediately characteristic subgroup		Yes

Properties related to transitivity of normality and conjugacy

Property	Meaning	Satisfied?	Explanation	Comment
hereditarily normal subgroup		Yes
transitively normal subgroup		Yes
conjugacy-closed subgroup		No
central factor		No

Properties related to position in lattice of subgroups

Property	Meaning	Satisfied?	Explanation	Comment
minimal normal subgroup	nontrivial normal subgroup not containing any other nontrivial normal subgroup	Yes		it is the unique minimal normal subgroup, hence also the socle
maximal normal subgroup	proper normal subgroup not contained in any other proper normal subgroup	Yes		it is the unique maximal normal subgroup, hence also the Jacobson radical
maximal subgroup	proper subgroup not contained in any other proper subgroup	Yes

GAP implementation

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.

Description	Functions used
SylowSubgroup(G,3)	SylowSubgroup
DerivedSubgroup(G)	DerivedSubgroup
FittingSubgroup(G)	FittingSubgroup
Socle(G)	Socle

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.