# Non-Hopf group

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 $x$ and $y$ and the single defining relation

$$x^{-1}y^2x=y^3.$$

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

#### References

 [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)