# 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?

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"Every epimorphism of $G$ is an automorphism" is unclear. It should be "Every epimorphism of $G$ onto itself is an automorphism." – Arturo Magidin Oct 31 '11 at 16:27
@Arturo - Right. I wanted to say epic endomorphism. – Chris Leary Oct 31 '11 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 $BS(2, 3)=\langle a, b; b^{-1}a^2b=a^3\rangle$.

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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) – user1729 Oct 31 '11 at 16:04
I think you mean page 197, not page 97. – Arturo Magidin Oct 31 '11 at 16:38
@user1729 - I don't speak German (or read it) either, which was greatly inhibiting my search of the literature. – Chris Leary Oct 31 '11 at 16:38
@ Arturo Magidin: Yes. Yes I do. Thanks! Too late to edit it though... – user1729 Oct 31 '11 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!).

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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.

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