S2 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:Z2 and the group is (up to isomorphism) symmetric group:S3 (see subgroup structure of symmetric group:S3).
VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group 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 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 $H_j$.

Subset in cycle decomposition notation	Subset in one-line notation	Description	Left coset of $H = H_3$?	Left coset of $H_1$?	Left coset of $H_2$?	Right coset of $H = H_3$?	Right coset of $H_1$?	Right coset of $H_2$?
$\{ (), (1,2) \}$	123 and 213	Fixes $\{ 3 \}$	Yes	No	No	Yes	No	No
$\{ (), (2,3) \}$	123 and 132	Fixes $\{ 1 \}$	No	Yes	No	No	Yes	No
$\{ (), (1,3) \}$	123 and 321	Fixes $\{ 2 \}$	No	No	Yes	No	No	Yes
$\{ (2,3), (1,3,2) \}$	132 and 312	Sends $3$ to $2$	Yes	No	No	No	No	Yes
$\{ (1,3), (1,2,3) \}$	321 and 231	Sends $3$ to $1$	Yes	No	No	No	Yes	No
$\{ (2,3), (1,2,3) \}$	132 and 231	Sends $2$ to $3$	No	No	Yes	Yes	No	No
$\{ (1,3), (1,3,2) \}$	321 and 312	Sends $1$ to $3$	No	Yes	No	Yes	No	No
$\{ (1,2), (1,2,3) \}$	213 and 231	Sends $1$ to $2$	No	Yes	No	No	No	Yes
$\{ (1,2), (1,3,2) \}$	213 and 312	Sends $2$ to $1$	No	No	Yes	No	Yes	No

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:

Subgroup: left/right	$\{ (), (1,2) \}$	$\{ (), (2,3) \}$	$\{ (), (1,3) \}$
$\{ (), (1,2) \}$	$\{ (), (1,2) \}$	$\{ (1,3), (1,2,3) \}$	$\{ (2,3), (1,3,2) \}$
$\{ (), (2,3) \}$	$\{ (1,3), (1,3,2) \}$	$\{ (), (2,3) \}$	$\{ (1,2), (1,2,3) \}$
$\{ (), (1,3) \}$	$\{ (2,3), (1,2,3) \}$	$\{ (1,2), (1,3,2) \}$	$\{ (), (1,3) \}$

Arithmetic functions

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

Effect of subgroup operators

In the table below, we provide values specific to $H$.

Function	Value as subgroup (descriptive)	Value as subgroup (link)	Value as group
normalizer	the subgroup itself	current page	cyclic group:Z2
centralizer	the subgroup itself	current page	cyclic group:Z2
normal core	trivial subgroup	--	trivial group
normal closure	whole group	--	symmetric group:S3
characteristic core	trivial subgroup	--	trivial group
characteristic closure	whole group	--	symmetric group:S3
commutator with whole group	subgroup $\{ (), (1,2,3), (1,3,2) \}$	A3 in S3	cyclic group:Z3

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 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.

Subgroup properties

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.

Property	Meaning	Satisfied?	Explanation	Comment
normal subgroup	equals all its conjugate subgroups	No	Distinct conjugates found above, see also S2 is not normal in S3	This is the smallest example (in order terms) of a group-subgroup pair where the subgroup is not normal in the whole group. It is also the smallest index example, since index two implies normal
characteristic subgroup	invariant under all automorphisms	No	(via normal)
coprime automorphism-invariant subgroup	invariant under all coprime automorphisms, i.e., automorphisms whose order is coprime to that of the group	Yes	The whole group is a complete group, so has no nontrivial coprime automorphisms
cofactorial automorphism-invariant subgroup	invariant under all cofactorial automorphisms, i.e., automorphisms whose order has no prime factors other than those in the group	No	(via normal)
subgroup-cofactorial automorphism-invariant subgroup	invariant under all automorphisms whose order has no prime factors other than those in the subgroup	No	not invariant under conjugation by $(1,3)$

Resemblance-based properties

Property	Meaning	Satisfied?	Explanation	Comment
Sylow subgroup	$p$-group of $p'$-index	Yes	2-Sylow subgroup
Hall subgroup	order and index are relatively prime	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)
order-automorphic subgroup		Yes
isomorph-conjugate subgroup		Yes
automorph-conjugate subgroup		Yes
pronormal subgroup		Yes	since it is a Sylow subgroup

Properties opposite to normality

Property	Meaning	Satisfied?	Explanation	Comment
abnormal subgroup		Yes	self-normalizing Sylow subgroup
weakly abnormal subgroup		Yes
self-normalizing subgroup	equals its own normalizer	Yes
self-centralizing subgroup	contains its centralizer in the whole group	Yes
core-free subgroup	normal core is trivial	Yes
contranormal subgroup	normal closure is whole group	Yes
maximal subgroup	proper subgroup not contained in any bigger proper subgroup	Yes

Linear representation theory

Further information: linear representation theory of symmetric group:S3, linear representation theory of cyclic group:Z2

Induced representations from subgroup to whole group

We consider induced representations from $H$ to $G$.

Representation of subgroup $H$	Induced representation on $G$ (in terms of character)	Irreducible components of induced representation on whole group
trivial representation	takes the value 3 at the identity element, 1 at each 2-transposition, and 0 at the 3-cycles	trivial representation and standard representation
sign representation	takes the value 3 at the identity element, -1 at each 2-transposition, and 0 at 3-cycles	sign representation and standard representation.

Restriction of representations to subgroups

Representation on whole group	Restriction (in terms of character)	Irreducible components of restriction
trivial representation	1 everywhere	trivial representation
sign representation	1 on identity, -1 on non-identity element	sign representation
standard representation	2 on identity, 0 on non-identity element	trivial representation and sign representation

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.

Representation/representation	Trivial	Sign
Trivial	1	0
Sign	0	1
Standard	1	1

GAP implementation

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]);;