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)

Comments

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

References

[a1]	G. Baumslag, D. Solitar, "Some two-generator one-relator non-Hopfian groups" Bull. Amer. Math. Soc. , 68 (1962) pp. 199–201