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).
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We consider the subgroup in the group defined as follows.
is the symmetric group of degree three, which, for concreteness, we take as the symmetric group on the set .
is the subgroup of comprising those permutations that fix . In particular, is the symmetric group on , embedded naturally in . It is isomorphic to cyclic group:Z2. As a set, contains two elements: and .
There are two other conjugate subgroups to in (so the total conjugacy class size of subgroups is 3). The two other subgroups are the subgroups fixing and respectively. Specifically, and its two other conjugate subgroups are:
With this notation, is the stabilizer of in the symmetric group on .
All these subgroups are -Sylow subgroups in .
See also subgroup structure of symmetric group:S3.
- 1 Cosets
- 2 Arithmetic functions
- 3 Effect of subgroup operators
- 4 Related subgroups
- 5 Subgroup properties
- 6 Linear representation theory
- 7 GAP implementation
There is a total of nine subsets of size two that arise as cosets of 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 . The coset parametrized by is the set of all elements that send to . This is a left coset of and a right coset of .
|Subset in cycle decomposition notation||Subset in one-line notation||Description||Left coset of ?||Left coset of ?||Left coset of ?||Right coset of ?||Right coset of ?||Right coset of ?|
|123 and 213||Fixes||Yes||No||No||Yes||No||No|
|123 and 132||Fixes||No||Yes||No||No||Yes||No|
|123 and 321||Fixes||No||No||Yes||No||No||Yes|
|132 and 312||Sends to||Yes||No||No||No||No||Yes|
|321 and 231||Sends to||Yes||No||No||No||Yes||No|
|132 and 231||Sends to||No||No||Yes||Yes||No||No|
|321 and 312||Sends to||No||Yes||No||Yes||No||No|
|213 and 231||Sends to||No||Yes||No||No||No||Yes|
|213 and 312||Sends to||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:
|order of whole group||6|
|order of subgroup||2|
|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 .
|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||A3 in S3||cyclic group:Z3|
The subgroup has prime index, hence is maximal, so there are no strictly intermediate subgroups between the subgroup and the whole group.
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 of order three, and intersects each coset of this subgroup.
Automorphisms and endomorphisms: properties satisfied
For ease of reference, we take here the subgroup , though the conclusions apply for the other two conjugates as well.
|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|
|Sylow subgroup||-group of -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)|
|pronormal subgroup||Yes||since it is a Sylow subgroup|
Properties opposite to normality
|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
Induced representations from subgroup to whole group
We consider induced representations from to .
|Representation of subgroup||Induced representation on (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.
Finding these subgroups inside a black-box symmetric group of degree three
gap> H := SylowSubgroup(G,2);; gap> L := AsList(ConjugacyClassSubgroups(G,H));; gap> H := L;;H1 := L;;H2 := L;;
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]);;