The Hopfian property for groups

Let $G$ be a group, which for my purposes would be abelian. To say that $G$ has the Hopf property is to say that every epimorphism of $G$ is an automorphism. Does anyone happen to recall the context in which Hopf first used this concept, and a reference for this?

• "Every epimorphism of $G$ is an automorphism" is unclear. It should be "Every epimorphism of $G$ onto itself is an automorphism." Oct 31, 2011 at 16:27
• @Arturo - Right. I wanted to say epic endomorphism. Oct 31, 2011 at 16:40

"H. Hopf, in 1932, raised the question as to whether a finitely generated group can be isomorphic to a proper factor of itself. This was answered in the affirmative: by B. H. Neumann, 1950, with a two-generator group with infinitely many defining relators; by G. Higman, 1951c, with a three-generator group with two defining relations; and by Baumslag and Solitar, 1962, with a two-generator group with one defining relator...A group which cannot be isomorphic to a proper factor of itself is called Hopfian."

-"Combinatorial Group Theory", Magnus, Karrass and Solitar (sec. 2.4)

The two-generator one-relator group given by Baumslag and Solitar is the imaginatively named Baumslag-Solitar group $BS(2, 3)=\langle a, b; b^{-1}a^2b=a^3\rangle$.

• Also, Magnus, Karrass and Solitar do not give a reference for a paper of Hopf from 1932. However, Lyndon and Schupp, in their book also called "Combinatorial Group Theory", say that this question first arose in a topological context, again by Hopf, in 1931. They cite the paper, "Beitrage zur Klassifizierung der Flachenabbildungen", J. Reine Angew. Math. 165, 225-236 (1931). But I don't speak German so I can't comment...(this is Chapter IV.4, p97 of my copy of Lyndon and Schupp) Oct 31, 2011 at 16:04
• I think you mean page 197, not page 97. Oct 31, 2011 at 16:38
• @user1729 - I don't speak German (or read it) either, which was greatly inhibiting my search of the literature. Oct 31, 2011 at 16:38
• @ Arturo Magidin: Yes. Yes I do. Thanks! Too late to edit it though... Oct 31, 2011 at 16:40

I have access to the first page of the paper by Hopf, but no more. Maybe it will provide enough context.

Here's the first paragraph:

Die Aufgabe, die Klassen der Abbildungen der geschlossenen orientierbaren Fläche $F_p$ vom Geschlecht $p$ auf die geschlossene orientierbare Fläche $F_q$ vom Geschlecht $q$ aufzuzählen, scheint mir sowohl wegen des Zusammenhanges mit funktionentheoretischen Fragen, - da eine über einer Riemannschen Fläche des Geschlechtes $q$ ausgebreitete Riemannsche Fläche des Geschlechtes $p$ eine derartige Abbildung definiert - als auch vom rein topologischen Standpunkt aus großen Interesses wert zu sein. Gelöst ist sie nur für spezielle $p$, $q$. Sieht man von diesen Sonderfällen, auf die wir sogleich zurückkommen werden, ab, so ist, wie man leight zeigt, die gewünschte Aufzählung identisch mit der Angabe aller Homomorphismen der Fundamentalgruppe $\mathfrak{G}_p$ von $F_p$ in die Fundamentalgrupper $\mathfrak{G}_q$ von $F_q$; aber dieses gruppentheoretische Problem dürfte kaum leichter zu erledigen sein als das ursprüngliche geometrische.

Now, running it through Google gives something a bit silly, but reading "between the lines", it seems to go something like this:

The task of enumerating the maps of a closed orientable surface $F_p$ of genus $p$ to a closed orientable surface $F_q$ of genus $q$, seems to me to be interesting because of the connection with function-theoretic issues - because a Riemann surface of genus p that has been extended from a Riemann surface of genus q defines such a map - and also from a purely topological point of view. It is solved only for special $p$ and $q$. Apart from these special case, to which we will come back, the desired list is equivalent to specifying all homomorphisms of the fundamental group $\mathfrak{G}_p$ of $F_p$ to the fundamental group $\mathfrak{G}_q$ of $F_q$; but this group-theoretical problem is unlikely to be easier to handle than the original geometric one.

Maybe someone with actual German speaking skills can fix the translation. I'm making this answer a Community wiki, so that should lower the bar for editing (feel free to do so!).

According to the Encyclopedia of Mathematics, the term derives from Heinz Hopf's question (in 1932) of whether there exist finitely generated non-Hopfian groups.