# Center of dihedral group:D8

From Groupprops

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) dihedral group:D8 (see subgroup structure of dihedral group:D8).

The subgroup is a normal subgroup and the quotient group is isomorphic to Klein four-group.VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part| Group-subgroup pairs with the same quotient part | All pages on particular subgroups in groups

This article discuss the diheral group of order eight and its center, which is a cyclic group of order two.

The dihedral group of order eight is defined as:

.

and the center is the cyclic subgroup .

## Contents

## Subgroup-defining functions yielding this subgroup=

- Center: The center of is . To see that every element of is in the center, note that commutes with both and . To see that no other element is in the center, note that and do not commute.
- Commutator subgroup: is the commutator subgroup. The quotient (called the abelianization) is , which is isomorphic to Klein four-group.
- Socle and monolith: In fact, this is the unique minimal normal subgroup.
- Frattini subgroup: is the intersection of the three maximal subgroups: .
- first agemo subgroup: . In other words, it is the subgroup generated by the squares.

Subgroup property | Meaning of subgroup property | Reason it is satisfied |
---|---|---|

normal subgroup | invariant under inner automorphisms | center is normal |

characteristic subgroup | invariant under all automorphisms | center is characteristic, commutator subgroup is characteristic |

fully invariant subgroup | invariant under all endomorphisms | commutator subgroup is fully invariant, agemo subgroups are fully invariant |

verbal subgroup | generated by set of words | commutator subgroup is verbal, agemo subgroups are verbal |

normal-isomorph-free subgroup | no other isomorphic normal subgroup |

Subgroup property | Meaning of subgroup property | Reason it is not satisfied |
---|---|---|

isomorph-free subgroup | No other isomorphic subgroups | There are other subgroups of order two. |

homomorph-containing subgroup | contains all homomorphic images | There are other subgroups of order two. |

Subgroup property | Meaning of subgroup property | Reason it is satisfied |
---|---|---|

central subgroup | contained in the center | In fact, it is equal to the center. |

central factor | (because it is central). | |

transitively normal subgroup | (because it is a central factor). | |

SCAB-subgroup |