Hopfian group

1. What is a hopfian group? 

   There are (at least) two equivalent definitions of a hopfian group.  The first one, which was written in the book "Topics in Combinatorial Group Theory" (Gilbert Baumslag) is shown below.  

   The second one, which was written in "Combinatorial Group Theory" (Roger Lyndon and Paul Schupp), is as follows: a group G is called hopfian if every homomorphism from G onto G is an isomorphism. The equivalence between these two definitions can be explained rigorously as follows:
   - If every homomorphism from G onto G is an isomorphism, for each normal subgroup N of G, consider the projection p : G -> G/N such that p(g) = gN for each g in G. If G is isomorphic to G/N, consider the isomorphism i : G/N -> G, it is obvious that the composition of p and i yields a homomorphism from G onto G. By our assumption, this composition must be an isomorphism and the projection is also an isomorphism, which means that N is the trivial group. 
   - Conversely, if for any proper normal subgroup N of G, G is not isomorphic to G/N, then for each homomorphism f from G onto G, Im(f) = G and is isomorphic to G/Ker(f). By our assumption, Ker(f) must be the trivial group, which means that f is an isomorphism. 
   Hopfian group was named after the German mathematician Heinz Hopf (1894 - 1971). 


                                                               Heinz Hopf (1894 - 1971)

2. Examples of hopfian groups and non-hopfian groups

   Example 1: Every finite group is hopfian. Proving this is almost equivalent to proving that any surjective mapping f from a finite set X to itself is also bijective. 
   Example 2: Every group G satisfying the ascending chain condition is hopfian. Indeed, if there is a nontrivial normal subgroup N of G such that G/N is isomorphic to G, we might construct an infinite ascending chain of subgroups as follows: 
   - Set G(0) = the trivial group, G(1) = N. 
   - For each n, we have G/G(n) is isomorphic to G via the isomorphism i : G -> G/G(n). Choose G(n + 1) such that i(G) = G(n + 1)/G(n). 
   This contradicts the definition of the ascending chain condition. 
   Example 3: Fix a prime number p. The multiplicative group of complex numbers z such that there exists a positive integer n satisfying z^(p^n) = 1 is not hopfian, since the mapping i sending each z to z^p is an onto homomorphism but not an isomorphism. This group is called Prüfer group, which was named after the German mathematician Heinz Prüfer.


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