×

Automorphisms fixing every normal subgroup of a nilpotent-by-Abelian group. (English) Zbl 1179.20034

If \(G\) is a group, the set \(\operatorname{Aut}_n(G)\) of all automorphisms fixing every normal subgroup of \(G\) is a subgroup of the full automorphism group \(\operatorname{Aut}(G)\) of \(G\), and \(\operatorname{Aut}_n(G)\) obviously contains the group \(\text{Inn}(G)\) consisting of all inner automorphisms of \(G\). It has been proved by S. Franciosi and the reviewer [Boll. Unione Mat. Ital., VII. Ser., B 1, 1161-1170 (1987; Zbl 0635.20014)] that if \(G\) is a nilpotent group, then \(\operatorname{Aut}_n(G)\) is nilpotent-by-Abelian; in the same paper it was also shown \(\operatorname{Aut}_n(G)\) is polycyclic when \(G\) is. In particular, if \(G\) is a finite soluble group, then \(\operatorname{Aut}_n(G)\) is soluble.
In the paper under review, the author proves that if the group \(G\) is (nilpotent of class \(c\))-by-Abelian, then \(\operatorname{Aut}_n(G)\) is (nilpotent of class \(c\))-by-metabelian; in partcular, if \(G\) is a metabelian group, then \(\operatorname{Aut}_n(G)\) is soluble with derived length at most \(3\). Among other results, it is also proved that if \(G\) is a supersoluble group, then \(\operatorname{Aut}_n(G)\) is finitely generated and nilpotent-by-(finite and supersoluble).

MSC:

20F28 Automorphism groups of groups
20E36 Automorphisms of infinite groups
20F19 Generalizations of solvable and nilpotent groups
20D45 Automorphisms of abstract finite groups

Citations:

Zbl 0635.20014
PDFBibTeX XMLCite
Full Text: DOI arXiv EuDML Link

References:

[1] C. D. H. COOPER, Power automorphisms of a group, Math. Z., 107 (1968), pp. 335-356. Zbl0169.33801 MR236253 · Zbl 0169.33801 · doi:10.1007/BF01110066
[2] G. ENDIMIONI, Pointwise inner automorphisms in a free nilpotent group, Quart. J. Math., 53 (2002), pp. 397-402. Zbl1060.20025 MR1949151 · Zbl 1060.20025 · doi:10.1093/qjmath/53.4.397
[3] G. ENDIMIONI, Normal automorphisms of a free metabelian nilpotent group, http://arxiv.org/abs/math.GR/0612347. Zblpre05656566 · Zbl 1187.20043
[4] S. FRANCIOSI and F. DE GIOVANNI, On automorphisms fixing normal subgroups of nilpotent groups, Boll. Un. Mat. Ital. B, 7 (1987), pp. 1161-1170. Zbl0635.20014 MR923446 · Zbl 0635.20014
[5] A. LUBOTZKY, Normal automorphisms of free groups, J. Algebra, 63 (1980), pp. 494-498. Zbl0432.20025 MR570726 · Zbl 0432.20025 · doi:10.1016/0021-8693(80)90086-1
[6] D. J. S. ROBINSON, A course in the theory of groups, Springer-Verlag, 1982. Zbl0483.20001 MR648604 · Zbl 0483.20001
[7] D. J. S. ROBINSON, Automorphisms fixing every subnormal subgroup of a finite group, Arch. Math., 64 (1995), pp. 1-4. Zbl0813.20021 MR1305651 · Zbl 0813.20021 · doi:10.1007/BF01193541
[8] D. SEGAL, Polycyclic groups, Cambridge University Press, 1983. Zbl0516.20001 MR713786 · Zbl 0516.20001
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.