# 2-Sylow subgroup of general linear group:GL(2,3)

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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) semidihedral group:SD16 and the group is (up to isomorphism) general linear group:GL(2,3) (see subgroup structure of general linear group:GL(2,3)).VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part | All pages on particular subgroups in groups

## Contents

## Definition

is the general linear group of degree two over field:F3. In other words, it is the group of invertible matrices with entries over the field of three elements. The field has elements with .

is one of the 2-Sylow subgroups of , i.e.:

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is isomorphic to semidihedral group:SD16.

## Arithmetic functions

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

order of the whole group | 48 | |

order of the subgroup | 16 | |

index of the subgroup | 3 | |

size of conjugacy class of subgroups | 3 | |

number of conjugacy classes in automorphism class | 1 |

## Effect of subgroup operators

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

normalizer | the subgroup itself | current page | semidihedral group:SD16 |

centralizer | center of general linear group:GL(2,3) | cyclic group:Z2 | |

normal core | 2-core of general linear group:GL(2,3) | quaternion group | |

normal closure | the whole group | -- | general linear group:GL(2,3) |

characteristic core | 2-core of general linear group:GL(2,3) | quaternion group | |

characteristic closure | the whole group | -- | general linear group:GL(2,3) |

commutator with whole group | all matrices of determinant 1 | SL(2,3) in GL(2,3) | special linear group:SL(2,3) |

## Conjugacy class-defining functions

Conjugacy class-defining function | Meaning in general | Why it takes this value |
---|---|---|

2-Sylow subgroups | -Sylow subgroups are subgroups whose order is a power of and index is relatively prime to . Sylow subgroups exist and Sylow implies order-conjugate, i.e., any two -Sylow subgroups are conjugate. Here . | The subgroups have order and index 3. |

2-Sylow normalizers | Normalizer of a 2-Sylow subgroup | The 2-Sylow subgroups are self-normalizing subgroups. |

## Subgroup properties

### Sylow and corollaries

The subgroup is a 3-Sylow subgroup, so many properties follow as a corollary of that.

Property | Meaning | Why being Sylow implies the property |
---|---|---|

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

isomorph-conjugate subgroup | conjugate to any subgroup isomorphic to it | (via order-conjugate) |

automorph-conjugate subgroup | conjugate to any subgroup automorphic to it | (via isomorph-conjugate) |

intermediately isomorph-conjugate subgroup | conjugate to any subgroup isomorphic to it inside any intermediate subgroup | Sylow implies intermediately isomorph-conjugate |

intermediately automorph-conjugate subgroup | automorph-conjugate in every intermediate subgroup | (via intermediately isomorph-conjugate) |

pronormal subgroup | conjugate to any conjugate subgroup in their join | Sylow implies pronormal |

weakly pronormal subgroup | (via pronormal) | |

paranormal subgroup | (via pronormal) | |

polynormal subgroup | (via pronormal) |

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

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

2-subnormal subgroup | normal subgroup in its normal closure | No | ||

subnormal subgroup | series from subgroup to whole group, each normal in next | No | ||

contranormal subgroup | normal closure is whole group | Yes | ||

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

abnormal subgroup | Yes | Sylow normalizer implies abnormal | ||

weakly abnormal subgroup | Yes | (via abnormal) | ||

self-centralizing subgroup | contains its centralizer in the whole group | Yes |