Hopf group

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A group that is not isomorphic to any of its proper quotient groups. The name was given in honour of H. Hopf, who raised in 1932 the question whether there exist finitely-generated groups (cf. Finitely-generated group) that do not have this property. Examples of non-Hopf groups are known, among them an example of a group with two generators and a single defining relation (cf. also Non-Hopf group). Every finitely-generated residually-finite group is a Hopf group.


[1] W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience) (1966)


A (non-) Hopf group is also called a (non-) Hopfian group.

How to Cite This Entry:
Hopf group. N.N. Vil'yams (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098