Non-Hopf group

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non-Hopfian group

A group that has an endomorphism onto itself with a non-trivial kernel, that is, a group that is isomorphic to a proper quotient group of itself. (Otherwise the group is called a Hopfian group, cf. Hopf group.) The term stems from Hopf's problem (1932) whether there are such groups that are finitely generated. It turned out that there are even finitely-presented non-Hopfian groups. An example of a finitely-generated non-Hopfian group is the group with two generators and and the single defining relation

Infinitely-generated non-Hopfian groups are quite easy to construct, for example, the direct product of infinitely many isomorphic groups.


[1] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[2] W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience) (1966)


The example of the two-generated non-Hopfian group mentioned above is due to G. Baumslag and D. Solitar [a1].


[a1] G. Baumslag, D. Solitar, "Some two-generator one-relator non-Hopfian groups" Bull. Amer. Math. Soc. , 68 (1962) pp. 199–201
How to Cite This Entry:
Non-Hopf group. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by le group','../s/s084370.htm','Simple group','../s/s085220.htm','Solvable group','../s/s086130.htm','Variety of groups','../v/v096290.htm','Wreath product','../w/w098160.htm')" style="background-color:yellow;">A.L. Shmel'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article