Abstract
A group G is an A-E group if the endomorphism nearring of G generated by its automorphisms equals the endomorphism nearring generated by its endomorphisms. In this paper we set out to determine those p-groups G that are semidirect products of cyclic groups and are A-E groups. We show that no such groups exist when p = 2. When p is odd, we show that G is an A-E group whenever the nilpotency class of G is less than p. Examples are given to show no conclusion can be drawn when the nilpotency class is greater than or equal to p.
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References
B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967
B. W. King, Presentations of metacyclic groups, Bull. Austral. Math. Soc., 8(1973), 103–131
C. G. Lyons and J. J. Malone, Finite dihedral groups and d. g. near rings II, Compositio Math., 26(1973), 249–259
C. G. Lyons and G. L. Peterson, Local endomorphism near-rings, Proc. Edinburgh Math. Soc, 31(1988), 409–414
C. J. Maxson, On local near-rings, Math. Z., 106(1968), 197–205
J. D. P. Meldrum, Near-Rings and their Links with Groups, Pitman, London, 1985
G. L. Peterson, Endomorphism near-rings of p-groups generated by the automorphism and inner automorphism groups, Proc. Amer. Soc., 119(1993), 1045–1047
G. L. Peterson, Finite metacyclic I-E and I-A groups, Comm. Alg., 23(1995), 4563–4585
G. L. Peterson, The semidirect products of finite cyclic groups that are I-E groups, Mh. Math., 121(1996), 275–290
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Peterson, G.L. Split Metacyclic p-Groups That Are A-E Groups. Results. Math. 36, 160–183 (1999). https://doi.org/10.1007/BF03322109
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DOI: https://doi.org/10.1007/BF03322109