# S3 in S4

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) symmetric group:S3 and the group is (up to isomorphism) symmetric group:S4 (see subgroup structure of symmetric group:S4).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 in the group defined as follows.

is the symmetric group of degree four, 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 symmetric group:S3. has order .

There are three other conjugate subgroups to in (so the total conjugacy class size of subgroups is 4). The other subgroups are the subgroups fixing , , and respectively.

The four conjugates are:

See also subgroup structure of symmetric group:S4.

## Cosets

There are four left cosets and four right cosets of each subgroup. Each left coset of a subgroup is a right coset of one of its conjugate subgroups. This gives a total of 16 subsets.

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 .

## Complements

There is a unique normal complement that is common to all the subgroups. This is the subgroup normal Klein four-subgroup of symmetric group:S4:

There is also a conjugacy class of subgroups each of which is a permutable complement to each of the s. These are cyclic four-subgroups of symmetric group:S4, and these are:

Note that the fact that these are permutable complements can be understood as a special case of Cayley's theorem. See also every group of given order is a permutable complement for symmetric groups, which says that any finite group of order is, via the Cayley embedding, a permutable complement to in .

Apart from these, each of the s has a number of lattice complements:

- Any subgroup generated by double transposition in S4 is a lattice complement to each in the whole group. Thus, each has three such lattice complements.
- For each , a subgroup of order three
*not*contained in that is a lattice complement to it. Thus, each has three such lattice complements, because one of the four subgroups of order three is contained in that .

Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|

retract | has a normal complement | Yes | subgroup above is a normal complement | |

permutably complemented subgroup | has a permutable complement | Yes | normal complement is permutable complement too | |

lattice-complemented subgroup | has a lattice complement | Yes | normal complement is lattice complement too | |

complemented normal subgroup | normal subgroup with permutable complement | No | not normal itself |

## Arithmetic functions

Function | Value | Explanation |
---|---|---|

order of whole group | 24 | |

order of subgroup | 6 | |

index of subgroup | 4 | |

size of conjugacy class of subgroup (=index of normalizer) | 4 | see above for list of conjugates |

number of conjugacy classes in automorphism class of subgroup | 1 | the whole group is a complete group, so the conjugation actions are precisely the automorphisms. |

size of automorphism class of subgroup | 4 | same as size of conjugacy class |