# Difference between revisions of "D8 in S4"

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<math>\! H_2 = \{ (), (1,2,4,3), (1,4)(2,3), (1,3,4,2), (1,2)(3,4), (1,3)(2,4), (1,4), (2,3) \}</math> | <math>\! H_2 = \{ (), (1,2,4,3), (1,4)(2,3), (1,3,4,2), (1,2)(3,4), (1,3)(2,4), (1,4), (2,3) \}</math> | ||

+ | |||

+ | ==Arithmetic functions== | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Function !! Value !! Explanation !! Comment | ||

+ | |- | ||

+ | | order of the group || 24 || || | ||

+ | |- | ||

+ | | order of the subgroup || 8 || || | ||

+ | |- | ||

+ | | [[index of a subgroup|index of the subgroup]] || [[arithmetic function value::index of a subgroup;3|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 <math>H</math>. | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Function !! Value as subgroup (descriptive) !! Value as subgroup (link) !! Value as group | ||

+ | |- | ||

+ | | [[normalizer]] || the subgroup itself || (current page) || [[dihedral group:D8]] | ||

+ | |- | ||

+ | | [[centralizer]] || <math>\{ (), (1,3)(2,4) \}</math> || [[subgroup generated by double transposition in S4]] || [[cyclic group:Z2]] | ||

+ | |- | ||

+ | | [[normal core]] || <math>\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}</math> || [[normal Klein four-subgroup of symmetric group:S4]] || [[Klein four-group]] | ||

+ | |- | ||

+ | | [[normal closure]] || the whole group || -- || [[symmetric group:S4]] | ||

+ | |- | ||

+ | | [[characteristic core]] || <math>\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}</math> || [[normal Klein four-subgroup of symmetric group:S4]] || [[Klein four-group]] | ||

+ | |- | ||

+ | | [[characteristic closure]] || the whole group || -- || [[symmetric group:S4]] | ||

+ | |} | ||

+ | |||

+ | ==Subgroup properties== | ||

+ | |||

+ | ===Resemblance-based properties=== | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Property !! Meaning !! Satisfied? !! Explanation !! Comment | ||

+ | |- | ||

+ | | [[satisfies property::order-conjugate subgroup]] || [[conjugate subgroups|conjugate]] to any subgroup of the same order || Yes || [[Sylow implies order-conjugate]] || | ||

+ | |- | ||

+ | | [[satisfies property::order-dominating subgroup]] || any subgroup of the whole group whose order divides the order of <math>H</math> is contained in a conjugate of <math>H</math> || Yes || [[Sylow implies order-dominating]] || | ||

+ | |- | ||

+ | | [[satisfies property::order-dominated subgroup]] || any subgroup of the whole group whose order is a multiple of the order of <math>H</math> contains a conjugate of <math>H</math> || Yes || [[Sylow implies order-dominated]] || | ||

+ | |- | ||

+ | | [[satisfies property::order-isomorphic subgroup]] || isomorphic to any subgroup of the group of the same order || Yes || (via order-conjugate) || | ||

+ | |- | ||

+ | | [[satisfies property::isomorph-automorphic subgroup]] || || Yes || (via order-conjugate) || | ||

+ | |- | ||

+ | | [[satisfies property::automorph-conjugate subgroup]] || || Yes || (via order-conjugate) || | ||

+ | |- | ||

+ | | [[satisfies property::order-automorphic subgroup]] || || Yes || (via order-conjugate) || | ||

+ | |- | ||

+ | | [[satisfies property::isomorph-conjugate subgroup]] || || Yes || (via order-conjugate) || | ||

+ | |} | ||

+ | |||

+ | ===Normality-related properties=== | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Property !! Meaning !! Satisfied? !! Explanation !! Comment | ||

+ | |- | ||

+ | | [[dissatisfies property::normal subgroup]] || equals all its [[conjugate subgroups]] || No || (see other conjugate subgroups) || | ||

+ | |- | ||

+ | | [[dissatisfies property::subnormal subgroup]] || || No || || | ||

+ | |- | ||

+ | | [[satisfies property::self-normalizing subgroup]] || equals its [[normalizer]] in the whole group || Yes || || | ||

+ | |- | ||

+ | | [[satisfies property::abnormal subgroup]] || || Yes || || | ||

+ | |- | ||

+ | | [[satisfies property::weakly abnormal subgroup]] || || Yes || || | ||

+ | |- | ||

+ | | [[satisfies property::contranormal subgroup]] || || Yes || || | ||

+ | |- | ||

+ | | [[satisfies property::maximal subgroup]] || || Yes || || | ||

+ | |- | ||

+ | | [[satisfies property::pronormal subgroup]] || || Yes || [[Sylow implies pronormal]] || | ||

+ | |- | ||

+ | | [[satisfies property::weakly pronormal subgroup]] || || Yes || (via pronormal) || | ||

+ | |} |

## Revision as of 23:02, 17 December 2010

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 has two other conjugate subgroups, which are given below:

and:

## Contents

## 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) |