# A3 in S3

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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.
VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part| Group-subgroup pairs with the same quotient part | All pages on particular subgroups in groups

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 $G$.

## 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.