# Subgroup generated by double transposition in symmetric group: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 generated by the double transposition . This is the permutation that interchanges with and with . Since the element has order two, is a two-element subgroup, isomorphic to cyclic group:Z2, and its two elements are the identity and .

There are two other conjugate subgroups to in (so the total conjugacy class size of subgroups is 3). The three subgroups are given below:

See also subgroup structure of symmetric group:S4.

## Contents

## Cosets

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## Complements

(and hence also each of its conjugate subgroups) has no permutable complements. However, it does have a lattice complement. Specifically, any S3 in S4 is a lattice complement to and also to each of its conjugates. For instance, .

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

retract | has a normal complement | No | ||

permutably complemented subgroup | has a permutable complement | No | ||

lattice-complemented subgroup | has a lattice complement | Yes |

## Subgroup properties

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

normal subgroup | equals all its conjugate subgroups | No | ||

2-subnormal subgroup | normal subgroup of normal subgroup | Yes | Normal in , which is the normal Klein four-subgroup of symmetric group:S4 | |

hypernormalized subgroup | taking normalizers repeatedly reaches the whole group | No | Normalizer is D8 in S4, which is self-normalizing | |

pronormal subgroup | any conjugate is conjugate to it in their join | No | pronormal and subnormal implies normal | |

contranormal subgroup | normal closure is whole group | No | ||

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

### Resemblance-based properties

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

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

isomorph-automorphic subgroup | automorphic to any subgroup isomorphic to it | No | The subgroup is isomorphic to it but not related to it via an automorphism | See S2 in S4 for this other automorphism class. |

automorph-conjugate subgroup | conjugate to any subgroup automorphic to it | Yes |