×

The group of outer automorphisms of an infinite nilpotent p-group and its normal p-subgroups. (Italian) Zbl 0588.20027

This paper is a contribution to the literature on normal p-subgroups of the automorphism groups of p-groups. The finite case was dealt with by Peter Schmid. Here it is shown that if G is an infinite nilpotent p-group and Out G has no non-identity normal p-subgroup, then there are three possibilities. (1) G is elementary Abelian. (2) G is divisible and p is odd. (3) G is the central product of P and C, where P is a finite extraspecial group of exponent \(p\neq 2\) and C is a quasicyclic p-group. The proof owes some details to a paper of Menegazzo and Stonehewer.
Reviewer: N.Blackburn

MSC:

20F28 Automorphism groups of groups
20F18 Nilpotent groups
20F50 Periodic groups; locally finite groups
20E07 Subgroup theorems; subgroup growth
20E36 Automorphisms of infinite groups
PDF BibTeX XML Cite
Full Text: Numdam EuDML

References:

[1] J. Buckley - J. Wiegold , On the number of outer automorphisms of an infinite nilpotent p-group , Arch. Math. , 31 ( 1978 ), pp. 321 - 328 . MR 522855 | Zbl 0387.20035 · Zbl 0387.20035
[2] W. Gaschutz , Nichtabelsche p-gruppen besitzen aüßere p-Automorphismen , J. Algebra , 4 ( 1966 ), pp. 1 - 2 . MR 193144 | Zbl 0142.26001 · Zbl 0142.26001
[3] B. Huppert , Endliche Gruppen I , Springer-Verlag , Berlin , Heidelberg , 1967 . MR 224703 | Zbl 0217.07201 · Zbl 0217.07201
[4] F. Menegazzo - S. Stonehewer , On the automorphism group of a nilpotent p-group , in corso di stampa presso il J. London Math. Soc. Zbl 0526.20023 · Zbl 0526.20023
[5] O. Müller , On p-automorphisms of finite p-groups , Arch. Math. , 32 ( 1979 ), pp. 533 - 538 . MR 550318 | Zbl 0417.20025 · Zbl 0417.20025
[6] D.J.S. Robinson , Finiteness conditions and generalized soluble groups, part 1 , Springer-Verlag , Berlin , Heidelberg , New York , 1972 . MR 332989 | Zbl 0243.20032 · Zbl 0243.20032
[7] P. Schmid , Normal p-subgroups in the group of outer automorphisms of a finite p-group , Math. Z. , 147 ( 1976 ), pp. 271 - 277 . Article | MR 419602 | Zbl 0315.20020 · Zbl 0315.20020
[8] A.E. Zalesski , A nilpotent p-group has an outer automorphism , Dokl. Akad. Nauk SSSR , 196 ( 1971 ) (traduzione in inglese: Soviet Math. Dokl. , 12 ( 1971 ), pp. 127 - 130 ). MR 274587 | Zbl 0226.20021 · Zbl 0226.20021
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.