# Cyclic four-subgroups of 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:Z4 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

This article is about a conjugacy class (also an equivalence class up to automorphisms) in the symmetric group of degree four, comprising cyclic groups of order four.

We denote the symmetric group on by . We define:

.

Its other images under inner automorphisms (which are the same as its images under automorphisms) are:

- .
- .

## Contents

## Arithmetic functions

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

order of whole group | 24 | |

order of subgroup | 4 | |

index | 6 | |

size of conjugacy class | 3 |

## Effect of subgroup operators

Function | Value as subgroup (descriptive) | value as subgroup (link) | Value as group |
---|---|---|---|

normalizer | D8 in S4 | dihedral group:D8 | |

centralizer | same as | cyclic group:Z4 | |

normal core | trivial subgroup | trivial group | |

normal closure | whole group | symmetric group:S4 |

## Related subgroups

### Intermediate subgroups

Value of intermediate subgroup (descriptive) | Isomorphism class of intermediate subgroup | Small subgroup in intermediate subgroup | Intermediate subgroup in big group |
---|---|---|---|

dihedral group:D8 | cyclic maximal subgroup of dihedral group:D8 | D8 in S4 |

### Smaller subgroups

Value of smaller subgroup (descriptive) | Isomorphism class of smaller subgroup | Smaller subgroup in subgroup | Smaller subgroup in whole group |
---|---|---|---|

cyclic group:Z2 | Z2 in Z4 |

### Isomorph-conjugate subgroup

`Further information: isomorph-conjugate subgroup, isomorph-automorphic subgroup, automorph-conjugate subgroup`

is an isomorph-conjugate subgroup of : it is conjugate to all the other subgroups isomorphic to it. Hence, it is also an isomorph-automorphic subgroup and an automorph-conjugate subgroup. Note that since is a complete group (symmetric groups are complete) *all* subgroups of are automorph-conjugate.

### Subgroup whose join with any distinct conjugate is the whole group

`Further information: subgroup whose join with any distinct conjugate is the whole group`

The join of and any conjugate of *distinct* from is the whole group. In particular, this forces that:

- is a pronormal subgroup and hence also a weakly pronormal subgroup.
- Since is an automorph-conjugate subgroup, this also forces that is a procharacteristic subgroup and weakly procharacteristic subgroup.

## Opposites of normality satisfied and dissatisfied

### Opposites satisfied

- is a core-free subgroup of : the normal core of in is trivial.
- is a contranormal subgroup of : the normal closure of in is .

### Opposites dissatisfied

- is
*not*a self-normalizing subgroup of . - is not an abnormal subgroup of or a weakly abnormal subgroup of , because both of these would imply that is a self-normalizing subgroup of .