S2 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:Z2 and the group is (up to isomorphism) symmetric group:S3 (see subgroup structure of symmetric group:S3).
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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 \}.

S3latticeofsubgroups.png

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]);;