# Difference between revisions of "D8 in S4"

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===Construction of subgroup given group as a black box=== | ===Construction of subgroup given group as a black box=== | ||

− | Suppose we are already given a group that we know to be isomorphic to [[symmetric group:S4]]. Then, the subgroup <math>H</math> can be constructed as follows: | + | Suppose we are already given a group that we know to be isomorphic to [[symmetric group:S4]]. Then, the subgroup <math>H</math> can be constructed using [[GAP:SylowSubgroup|SylowSubgroup]] as follows: |

<tt>H := SylowSubgroup(G,2);</tt> | <tt>H := SylowSubgroup(G,2);</tt> |

## Revision as of 01:02, 23 May 2012

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) dihedral group:D8 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 the subgroup in the group , where is symmetric group:S4, i.e., the symmetric group on the set , and is the subgroup:

is a 2-Sylow subgroup of and is isomorphic to dihedral group:D8. It has two other conjugate subgroups, which are given below:

and:

## Contents

## Complements

The permutable complements to (and also to each of its conjugates) are precisely the subgroups of order three, namely A3 in S4 and its conjugates. Each of these is a conjugate to each of the conjugates of :

In addition, any S2 in S4 is a lattice complement but *not* a permutable complement to two of the three conjugates of , namely the ones that do not contain it.

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

permutably complemented subgroup | has a permutable complement | Yes | A3 in S4 | |

lattice-complemented subgroup | has a lattice complement | Yes | A3 in S4, or S2 in S4 | |

retract | has a normal complement | No | ||

complemented normal subgroup | No | not a normal subgroup |

## Arithmetic functions

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

order of the group | 24 | ||

order of the subgroup | 8 | ||

index of the subgroup | 3 | ||

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) | dihedral group:D8 |

centralizer | subgroup generated by double transposition in S4 | cyclic group:Z2 | |

normal core | normal Klein four-subgroup of symmetric group:S4 | Klein four-group | |

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

characteristic core | normal Klein four-subgroup of symmetric group:S4 | Klein four-group | |

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

## Subgroup properties

### Resemblance-based properties

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

order-conjugate subgroup | conjugate to any subgroup of the same order | Yes | Sylow implies order-conjugate | |

order-dominating subgroup | any subgroup of the whole group whose order divides the order of is contained in a conjugate of | Yes | Sylow implies order-dominating | |

order-dominated subgroup | any subgroup of the whole group whose order is a multiple of the order of contains a conjugate of | Yes | Sylow implies order-dominated | |

order-isomorphic subgroup | isomorphic to any subgroup of the group of the same order | Yes | (via order-conjugate) | |

isomorph-automorphic subgroup | Yes | (via order-conjugate) | ||

automorph-conjugate subgroup | Yes | (via order-conjugate) | ||

order-automorphic subgroup | Yes | (via order-conjugate) | ||

isomorph-conjugate subgroup | Yes | (via order-conjugate) |

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

normal subgroup | equals all its conjugate subgroups | No | (see other conjugate subgroups) | |

subnormal subgroup | No | |||

self-normalizing subgroup | equals its normalizer in the whole group | Yes | ||

abnormal subgroup | Yes | |||

weakly abnormal subgroup | Yes | |||

contranormal subgroup | Yes | |||

maximal subgroup | Yes | |||

pronormal subgroup | Yes | Sylow implies pronormal | ||

weakly pronormal subgroup | Yes | (via pronormal) |

## Fusion system on subgroup

The subgroup embedding induces the non-inner non-simple fusion system for dihedral group:D8.

## GAP implementation

### Construction of subgroup given group as a black box

Suppose we are already given a group that we know to be isomorphic to symmetric group:S4. Then, the subgroup can be constructed using SylowSubgroup as follows:

`H := SylowSubgroup(G,2);`

### Construction of group-subgroup pair

The group and subgroup pair can be defined using GAP's SymmetricGroup and SylowSubgroup functions as follows:

`G := SymmetricGroup(4); H := SylowSubgroup(G,2);`

Note that this doesn't output the subgroup used on this page, but rather the conjugate subgroup . However all relevant properties are invariant under conjugation, so this does not matter.