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单词 HopfianGroup
释义

Hopfian group


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finite rank

A group is said to be Hopfian if it is not isomorphicPlanetmathPlanetmathPlanetmath to any of its proper quotients (http://planetmath.org/QuotientGroup).A group G is Hopfian if and only if every surjective endomorphismMathworldPlanetmathPlanetmath GG is an automorphism.

A group is said to be co-Hopfian if it is not isomorphic to any of its proper subgroupsMathworldPlanetmath.A group G is co-Hopfian if and only if every injective endomorphism GG is an automorphism.

Examples

Every finite groupMathworldPlanetmath is obviously Hopfian and co-Hopfian.

The group of rationals is an example of an infinite group that is both Hopfian and co-Hopfian.

The group of integers is Hopfian, but not co-Hopfian.More generally, every finitely generatedMathworldPlanetmathPlanetmathPlanetmath abelian groupMathworldPlanetmath is Hopfian,but is not co-Hopfian unless it is finite.

Quasicyclic groups are co-Hopfian, but not Hopfian.

Free groupsMathworldPlanetmath of infinite rank are neither Hopfian nor co-Hopfian.By contrast, free groups of finite rank are Hopfian (though not co-Hopfian unless of rank zero).

By a theorem of Mal’cev, every finitely generated residually finite group is Hopfian.

The Baumslag-Solitar group with presentationMathworldPlanetmathPlanetmathPlanetmath b,tt-1b2t=b3 is an example of a finitely generated group that is not Hopfian.

TitleHopfian group
Canonical nameHopfianGroup
Date of creation2013-03-22 15:36:03
Last modified on2013-03-22 15:36:03
Owneryark (2760)
Last modified byyark (2760)
Numerical id11
Authoryark (2760)
Entry typeDefinition
Classificationmsc 20F99
Related topicHopfianModule
DefinesHopfian
Definesco-Hopfian
Definescohopfian
Definesco-Hopfian group
Definescohopfian group
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