×

ZS-metacyclic groups and their endomorphism near-rings. (English) Zbl 0820.16043

The first author and C. G. Lyons started the study of endomorphism near-rings [in Proc. Edinb. Math. Soc., II. Ser. 17, 71-78 (1970; Zbl 0203.336)]. This paper follows in that line and determines the endomorphism near-rings of a large class of groups, using an extension of the methods developed in the paper above.
Let \(S\) be a semigroup of endomorphisms of the group \(G\). Then \(S\) generates a near-ring of mappings from \(G\) to itself. The three important cases are \(S = \text{Inn }G\), the inner automorphisms, \(S = \text{Aut }G\), all the automorphisms, \(S = \text{End }G\), all the endomorphisms. The corresponding nearrings are called \(I(G)\), \(A(G)\) and \(E(G)\) respectively. The groups considered here are ZS-metacyclic groups, which are extensions of a cyclic group by another cyclic group of order prime to the first. This is equivalent to saying that all the Sylow subgroups are cyclic.
The authors determine all the endomorphisms of a ZS-metacyclic group \(G\). They use a Peirce decomposition method to obtain the structure of \(I(G)\). Then they show that \(I(G) = E(G)\), so every endomorphism of \(G\) is a sum of inner automorphisms. This is a major advance in the study of \(I\)-\(E\) groups, groups \(H\) for which \(I(H) = E(H)\). Finally the radical of \(I(G)\) is determined. As may be expected, there is a great deal of careful detailed work in this paper.

MSC:

16Y30 Near-rings
20D45 Automorphisms of abstract finite groups
16S50 Endomorphism rings; matrix rings
20M20 Semigroups of transformations, relations, partitions, etc.
16N80 General radicals and associative rings

Citations:

Zbl 0203.336

Software:

Maple
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Beidleman, J.: Distributively generated near-rings with descending chain condition. Math. Z.91, 65-69 (1966). · Zbl 0131.01604
[2] Carmichael, R. D.: Introduction to the Theory of Groups of Finite Order. New York: Ginn. 1937. · Zbl 0019.19702
[3] Char, B., Geddes, K., Gonnet, G., Leong, B., Monagan, M., Watt, S., Maple V. Language Reference Manual, New York: Springer. 1991. · Zbl 0758.68038
[4] Coxeter, H. S. M., Moser, W. O. J.: Generators and Relations for Discrete Groups, 4th edn. Berlin: Springer. 1980. · Zbl 0422.20001
[5] Frohlich, A.: The near-rings generated by the inner automorphisms of a finite simple group. J. Lond. Math. Soc.33, 95-197 (1958). · Zbl 0084.26202
[6] Johnson, M.: Radicals of endomorphism near-rings. Rocky Mtn. J.3, 1-7 (1973). · Zbl 0253.16025
[7] Lyons, C. G., Mason, G.: Endomorphism near-rings of dicyclic and generalized dihedral groups. Proc. Roy. Ir. Acad.91A, 99-111 (1991). · Zbl 0713.16025
[8] Malone, J.J., Lyons, C.G.: Endomorphism near-rings, Proc. Edin. Math. Soc.17, 71-78 (1970). · Zbl 0203.33601
[9] Malone, J.J., Lyons, C.G.: Finite dihedral groups and d.g. near-rings I. Composito Math.24, 305-312 (1972). · Zbl 0236.16027
[10] Malone, J.J., Lyons, C.G.: Finite dihedral groups and d.g. near-rings II. Composito Math.26, 249-259 (1973). · Zbl 0266.16034
[11] Meldrum, J.: On the structure of morphism near-rings. Proc. Roy. Soc. Edin.81A, 287-298 (1978). · Zbl 0415.16026
[12] Meldrum, J.: Near-Rings and Their Links with Groups. Boston: Pitman. 1985. · Zbl 0658.16029
[13] Peterson, G.: On the structure of an endomorphism near-ring. Proc. Edin. Math. Soc.32, 223-229 (1989). · Zbl 0648.16029
[14] Pilz, G.: Near-Rings 2nd edn. Amsterdam: North Holland. 1983. · Zbl 0521.16028
[15] Scott, W.R.: Group Theory. New Jersey: Prentice-Hall. 1964. · Zbl 0126.04504
[16] Thomas, A., Wood, G.: Group Tables. Exter: Shiva. 1980.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.