# S2 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) cyclic group:Z2 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 pointwise. 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 five other conjugate subgroups to in (so the total conjugacy class size of subgroups is 3). Each subgroup fixed pointwise a subset of size two. Equivalently, each subgroup comprises the identity element and a 2-transposition. Specifically, and its two other conjugate subgroups are:

See also subgroup structure of symmetric group:S4.

## Contents

## Cosets

Each of these subgroups has 12 left cosets and 12 right cosets. Further, every left coset of one subgroup is a right coset of one of its conjugate subgroups. Overall, there are thus 72 cosets. Each coset is characterized by a fixed behavior on two of the four points.

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## Complements

There is a unique permutable complement to all these subgroups, which is also a normal subgroup and hence a normal complement. In particular, each of the subgroups is a retract. The unique permutable complement is A4 in S4, i.e., the alternating group of degree four.

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

retract | has a normal complement | Yes | alternating group:A4 (see A4 in S4) is a complement. | |

permutably complemented subgroup | has a permutable complement | Yes | (via retract) | |

lattice-complemented subgroup | has a lattice complement | Yes | (via retract) | |

direct factor | normal subgroup with normal complement | No | not normal | |

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

## Arithmetic functions

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

order of whole group | 24 | |

order of subgroup | 2 | |

index of the subgroup | 12 | |

size of conjugacy class | 6 | |

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 | non-normal Klein four-subgroups of symmetric group:S4 | Klein four-group | |

centralizer | non-normal Klein four-subgroups of symmetric group:S4 | Klein four-group | |

normal core | trivial subgroup | trivial | trivial group |

normal closure | whole group | -- | symmetric group:S4 |

characteristic core | trivial subgroup | trivial | trivial group |

characteristic closure | whole group | -- | symmetric group:S4 |

## Related subgroups

### Intermediate subgroups

The values given here are specific to .

Value of intermediate subgroup (descriptive) | Isomorphism class of intermediate subgroup | Number of conjugacy classes of intermediate subgroup fixing subgroup and whole group | Subgroup in intermediate subgroup | Intermediate subgroup in whole group |
---|---|---|---|---|

Klein four-group | 1 | Z2 in V4 | non-normal Klein four-subgroups of symmetric group:S4 | |

symmetric group:S3 | 3 | S2 in S3 | S3 in S4 | |

dihedral group:D8 | 1 | non-normal subgroups of dihedral group:D8 | D8 in S4 |

### Smaller subgroups

The subgroup has order two, hence is minimal, and has no smaller nontrivial subgroups.

## Subgroup properties

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

normal subgroup | equals all its conjugate subgroups | No | See list of conjugate subgroups | |

2-subnormal subgroup | normal in its normal closure | No | Normal closure is whole group | |

subnormal subgroup | there is a series from the subgroup to the whole group, each normal in the next | No | ||

contranormal subgroup | normal closure is the whole group | Yes | Normal closure is whole group | transpositions generate the finitary symmetric group |

self-normalizing subgroup | equals its normalizer in the whole group | No | Normalizer is | |

hypernormalized subgroup | taking normalizer repeatedly reaches the whole group | No | Normalizer is self-normalizing | |

pronormal subgroup | any conjugate subgroup to the subgroup is conjugate to it in their join | No | The subgroup is a conjugate, but in their join , which is abelian, the two subgroups are not conjugates. | |

weakly pronormal subgroup | No | |||

paranormal subgroup | No | |||

polynormal subgroup | No | |||

weakly normal subgroup | No |

### Resemblance-based properties

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

order-isomorphic subgroup | any subgroup of the whole group of the same order is isomorphic to it | Yes | any subgroup of order two is isomorphic to cyclic group:Z2 | |

isomorph-automorphic subgroup | any subgroup of the whole group isomorphic to it is related to it by an automorphism | No | The subgroup is isomorphic but there is no automorphism sending to this subgroup. | This other subgroup is subgroup generated by double transposition in symmetric group:S4. |

automorph-conjugate subgroup | any subgroup automorphic to it is conjugate to it | Yes | The whole group is a complete group -- all automorphisms of it are inner automorphisms. | See symmetric groups are complete |