View source for 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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067060/n0670601.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067060/n0670602.png" /> and the single defining relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067060/n0670603.png" /></td> </tr></table>

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,   "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Magnus,   A. Karrass,   B. Solitar,   "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience)  (1966)</TD></TR></table>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Baumslag,   D. Solitar,   "Some two-generator one-relator non-Hopfian groups"  ''Bull. Amer. Math. Soc.'' , '''68'''  (1962)  pp. 199–201</TD></TR></table>