Hopfian group

• Every finite group, by an elementary counting argument.
• More generally, every polycyclic-by-finite group.
• Any finitely generated free group.
• The additive group Q of rationals.
• Any finitely generated residually finite group.
• Any word-hyperbolic group.

• Quasicyclic groups.
• The additive group R of real numbers.{{cite web

| last = Clark

| first = Pete L.

| date = Feb 17, 2012

| url = https://math.stackexchange.com/a/110248/

| title = Can you always find a surjective endomorphism of groups such that it is not injective?

| website = Math Stack Exchange

| quote = This is because (R,+) is torsion-free and divisible and thus a Q-vector space. So -- since every vector space has a basis, by the Axiom of Choice -- it is isomorphic to the direct sum of copies of (Q,+) indexed by a set of continuum cardinality. This makes the Hopfian property clear.

}}

• The Baumslag–Solitar group B(2,3). (In general B(m, n) is non-Hopfian if and only if there exists primes p, q with p|m, q|n and p ∤ n, q ∤ m){{cite book

|title = Presentation of Groups

|author = Florian Bouyer

|publisher = University of Warwick

|url = https://warwick.ac.uk/fac/sci/maths/people/staff/fbouyer/presentation_of_group.pdf

|section = Theorem 7.7.

}}

It was shown by {{harvtxt|Collins|1969}} that it is an undecidable problem to determine, given a finite presentation of a group, whether the group is Hopfian. Unlike the undecidability of many properties of groups this is not a consequence of the Adian–Rabin theorem, because Hopficity is not a Markov property, as was shown by {{harvtxt|Miller|Schupp|1971}}.

{{reflist}}

• {{Cite journal | doi = 10.1007/BF01899291| title = On recognising Hopf groups| journal = Archiv der Mathematik| volume = 20| issue = 3| pages = 235–240| year = 1969| last1 = Collins | first1 = D. J. | s2cid = 119354919}}
• {{cite book | first=D. L.|last= Johnson | title=Presentations of groups | series=London Mathematical Society Student Texts | volume=15 | publisher=Cambridge University Press | year=1990 | isbn=0-521-37203-8 | page=35 }}
• {{Cite journal | doi = 10.1016/0021-8693(71)90028-7| title = Embeddings into hopfian groups| journal = Journal of Algebra| volume = 17| issue = 2| pages = 171| year = 1971| last1 = Miller | first1 = C. F. | last2 = Schupp | first2 = P. E. |author2link = Paul Schupp| doi-access = }}

• {{PlanetMath |urlname=HopfianGroup |title=Hopfian group}}
• [https://encyclopediaofmath.org/wiki/Non-Hopf_group Non-Hopf group] in the Encyclopedia of Mathematics

Category:Infinite group theory

Category:Properties of groups

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