On nilpotent groups with close groups of automorphisms

In a previous paper [7], I have made a study of the ”nilpotent” analogue of Hurwitz theorem [4] by considering a particular family of signatures called ”nilpotent admissible” [5]. We saw however, that if μN(g) represents the order of the largest nilpotent group of automorphisms of a surface of genus g < 2, then μN(g) < 16(g − 1) and this upper bound occurs when the covering group is a triangle group having the signature (0; 2,4,8) which is in its own 2-local formThe restriction to the nilpotent groups enabled me to obtain much more precise information than was available in the general case. Moreover, all nilpotent groups attaining this maximum order turned out to be ”2-groups”. Since every finite nilpotent group is the direct product of its Sylow subgroups and the groups of automorphisms are factor groups of the Fuchsian groups, it is natural for us to study the Fuchsian groups havin p-local signatures to obtain more precise information about the finite p-groups, and hence about the finite nilpotent groups.This suggests a new problem of determining for each prime p, the “p-group” analogue of Hurwitz theorem. It turns out, as often happens in questions of this nature, that p = 2 and p = 3 are indeed quite exceptional and harder to deal with while computing their lower central series than other primes. Actually, p = 3 is the most difficult, but all the other primes p ≥ 5 can be dealt with at once.

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REGULAR-NILPOTENT GROUPS OF AUTOMORPHISMS

Glasgow Mathematical Journal ◽

10.1017/s0017089502008923 ◽

2003 ◽

Vol 45 (1) ◽

pp. 1-5

Author(s):

REZA ZOMORRODIAN

Keyword(s):

Nilpotent Groups ◽

Groups Of Automorphisms

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§34. Nilpotent groups of automorphisms

de Gruyter Expositions in Mathematics - Groups of Prime Power Order, Volume 1 ◽

10.1515/9783110208221.318 ◽

2009 ◽

Keyword(s):

Nilpotent Groups ◽

Groups Of Automorphisms

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Supersoluble groups of automorphisms of compact Riemann surfaces

Glasgow Mathematical Journal ◽

10.1017/s0017089500007886 ◽

1989 ◽

Vol 31 (3) ◽

pp. 321-327 ◽

Cited By ~ 4

Author(s):

Grzegorz Gromadzki ◽

Colin MacLachlan

Keyword(s):

Finite Groups ◽

Riemann Surfaces ◽

Nilpotent Groups ◽

Metabelian Groups ◽

Supersoluble Groups ◽

Soluble Groups ◽

Cyclic Groups ◽

Groups Of Automorphisms ◽

Compact Riemann Surfaces ◽

Infinite Sequences

Given an integer g ≥ 2 and a class of finite groups let N(g, ) denote the order of the largest group in that a compact Riemann surface of genus g admits as a group of automorphisms. For the classes of all finite groups, cyclic groups, abelian groups, nilpotent groups, p-groups (given p), soluble groups and finally for metabelian groups, an upper bound for N(g, ) as well as infinite sequences for g for which this bound is attained were found in [5, 6, 7, 8, 13], [4], [10], [15], [16], [1], [2] respectively. This paper deals with that problem for the class of finite supersoluble groups i.e. groups with an invariant series all of whose factors are cyclic. In addition, it goes further by describing exactly those values of g for which the bound is attained. More precisely we prove:

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Automorphisms of finite order of nilpotent groups II

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.51.2014.4.1297 ◽

2014 ◽

Vol 51 (4) ◽

pp. 547-555 ◽

Cited By ~ 1

Author(s):

B. Wehrfritz

Keyword(s):

Nilpotent Group ◽

Finite Order ◽

Finite Index ◽

Nilpotent Groups ◽

Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

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