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# Property:Quick phrase

From Groupprops

## Pages using the property "Quick phrase"

Showing 25 pages using this property.

C | |
---|---|

Central factor + | factor in central product +, product with centralizer is whole group +, quotient action by outer automorphisms is trivial +, … |

Characteristic closure + | smallest characteristic subgroup containing +, intersection of all characteristic subgroups containing +, join of all automorphic subgroups + |

Characteristic subgroup + | invariant under all automorphisms +, automorphism-invariant +, strongly normal +, … |

Closure-characteristic subgroup + | normal closure is characteristic +, join of all conjugates is characteristic + |

Core-characteristic subgroup + | intersection of all conjugates is characteristic +, normal core is characteristic + |

Cosimplicial group + | cosimplicial object in category of groups +, group object in category of cosimplicial sets + |

Countable group + | countable +, countably generated + |

D | |

Dedekind group + | every subgroup is normal +, every cyclic subgroup is normal +, normal closure of element is cyclic + |

Direct factor + | factor in internal direct product +, normal with normal complement +, has centralizing complement + |

F | |

Fully invariant subgroup + | invariant under all endomorphisms +, endomorphism-invariant + |

G | |

Group of nilpotency class two + | class two +, inner automorphism group is abelian +, commutator subgroup inside center +, … |

Group whose automorphism group is abelian + | abelian automorphism group +, any two automorphisms commute + |

Gyrogroup + | left identity, left inverses and associativity twisted by an automorphism + |

H | |

H-group + | topological space with continuous operations that satisfy homotopy versions of group laws + |

Hypergroup + | variation of group where the multiplication operation is multi-valued + |

I | |

Injective endomorphism-invariant subgroup + | invariant under all injective endomorphisms +, injective endomorphism-invariant + |

Inverse semigroup + | semigroup where every element has a unique ''inverse'' in the semigroup sense (i.e., without reference to neutral elements) + |

Isomorph-containing subgroup + | contains all isomorphic subgroups +, weakly closed in any ambient group + |

Isomorph-free subgroup + | no other isomorphic subgroups +, no isomorphic copies +, only subgroup of its isomorphism type + |

L | |

Lie ring of nilpotency class two + | class two +, inner derivation Lie ring is abelian +, derived subring inside center +, … |

Loop + | group without associativity +, unital magma with unique left and right quotients + |

M | |

Metabelian group + | abelian-by-abelian +, abelian normal subgroup with abelian quotient +, abelian derived subgroup +, … |

Monoid + | group without inverses +, set with associative binary operation with identity element +, semigroup with identity element (neutral element) + |

Moufang loop + | loop (identity, inverses, not necessarily associative) with some associativity-like conditions that come close to making it a group + |

Multiary group + | group-like structure with n-ary instead of binary operation for multiplication + |