A group G is said to be hopfian
if every surjective endomorphism of G is an automorphism of G. Some authors have
investigated the problem of forming new hopfian groups in some familiar way from
given hopfian groups. This investigation is continued in this paper. Section 1
contains a statement of our main results and a discussion of these results.
Sections 2 and 3 contain the proof of these results. We list our main results
below.
Let the group G be a semidirect product of its subgroups H and F; that is,
H△G,G = HF and H ∩ F = 1. If H is a hopfian abelian group, we show that G is
hopfian if either of the following holds:
(a) F obeys the maximal condition for subgroups and H does not have an infinite
cyclic direct factor.
(b) F is a free abelian group of finite rank. Let H be a hopfian abelian normal
subgroup of G (which is not necessarily a split extension of H). Suppose G∕H
satisfies the maximal condition for subgroups. Then G is hopfian if any of the
following holds:
(c) H is a torsion group.
(d) H is of finite rank and has a hopfian torsion group.
(e) G = H ⋅F where F is a finite group. Let A be a hopfian group and let A×B
be the direct product of A and B. We will show A × B is hopfian if either of the
following holds:
(f) B is a finite solvable group with exactly n proper normal subgroups which
form a chain.
(g) B is a finite group of cube free order.
(h) B is a finite group of order p^{3},p a prime.
Finally we give some conditions on Z(A), the center of A, which will guarantee
the hopficity of A × B if B is an infinite cyclic group.
