Hopfian group: Difference between revisions
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Revision as of 23:49, 4 May 2010
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
This is a variation of finite group|Find other variations of finite group |
This property makes sense for infinite groups. For finite groups, it is always true
Definition
Symbol-free definition
A group is termed Hopfian if it satisfies the following equivalent conditions:
- It is not isomorphic to the quotient group by any nontrivial normal subgroup (in short, it is not isomorphic to any of its proper quotients).
- Every surjective endomorphism of it is an automorphism
Definition with symbols
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Finite group | order is finite | |FULL LIST, MORE INFO | ||
| Finitely generated free group | free group on finite generating set | finitely generated and free implies Hopfian | |FULL LIST, MORE INFO | |
| Finitely generated residually finite group | finitely generated and residually finite | finitely generated and residually finite implies Hopfian | |FULL LIST, MORE INFO | |
| Group satisfying ascending chain condition on normal subgroups | there is no infinite strictly ascending chain of normal subgroups | ascending chain condition on normal subgroups implies Hopfian | |FULL LIST, MORE INFO | |
| Slender group | every subgroup is finitely generated; equivalently, no strictly ascending chain of subgroups | slender implies Hopfian | |FULL LIST, MORE INFO |