Hopfian group
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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 finiteness (groups)|Find other variations of finiteness (groups) |
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
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