Vol. 32, No. 3, 1970

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On hopfian groups

R. Hirshon

Vol. 32 (1970), No. 3, 753–766

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 semi-direct product of its subgroups H and F; that is, HG,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 p3,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.

Mathematical Subject Classification
Primary: 20.27
Received: 11 October 1968
Revised: 8 September 1969
Published: 1 March 1970
R. Hirshon