Classification of p-groups of automorphisms of Riemann surfaces and their lower central series

1987 ◽  
Vol 29 (2) ◽  
pp. 237-244 ◽  
Author(s):  
Reza Zomorrodian
Keyword(s):  
Nilpotent Group ◽  
Riemann Surfaces ◽  
Lower Central Series ◽  
Nilpotent Groups ◽  
Fuchsian Groups ◽  
Central Series ◽  
Local Form ◽  
Precise Information ◽  
Groups Of Automorphisms ◽  
Hurwitz Theorem

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.

scholarly journals Lower central depth in finitely generated soluble-by-finite groups

1978 ◽  
Vol 19 (2) ◽  
pp. 153-154 ◽  
Author(s):  
John C. Lennox
Keyword(s):  
Finite Number ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Finite Depth ◽  
Lower Central Series ◽  
Central Series ◽  
Finitely Generated ◽  
Central Depth

We say that a group G has finite lower central depth (or simply, finite depth) if the lower central series of G stabilises after a finite number of steps.In [1], we proved that if G is a finitely generated soluble group in which each two generator subgroup has finite depth then G is a finite-by-nilpotent group. Here, in answer to a question of R. Baer, we prove the following stronger version of this result.

Finite-by-nilpotent groups

1956 ◽  
Vol 52 (4) ◽  
pp. 611-616 ◽  
Author(s):  
P. Hall
Keyword(s):  
Commutator Subgroup ◽  
Lower Central Series ◽  
Nilpotent Groups ◽  
Central Series ◽  
Image Position

1. Introduction. 1·1. Notation. Letandbe, respectively, the upper and lower central series of a group G. By definition, Zi+1/Zi is the centre of G/Zi and Γj+1 = [Γj, G] is the commutator subgroup of Γj with G. When necessary for clearness, we write ZiG) for Zi and Γj(G) for Γj.

scholarly journals Longer nilpotent series for classical unipotent subgroups

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Joshua Maglione
Keyword(s):  
Lower Central Series ◽  
Nilpotent Groups ◽  
Central Series

AbstractIn studying nilpotent groups, the lower central series and other variations can be used to construct an associated ℤ

On Nilpotent Products of Cyclic Groups—Reexamined by the Commutator Calculus

1975 ◽  
Vol 27 (6) ◽  
pp. 1185-1210 ◽  
Author(s):  
Hermann V. Waldinger ◽  
Anthony M. Gaglione
Keyword(s):  
Finite Number ◽  
Nilpotent Group ◽  
Free Product ◽  
Infinite Order ◽  
Lower Central Series ◽  
Central Series ◽  
Cyclic Groups ◽  
Collection Process

Ruth R. Struik investigated the nilpotent group , where G is a free product of a finite number of cyclic groups, not all of which are of infinite order, and Gm is the mth subgroup of the lower central series of G. Making use of the “collection process” first given by Philip Hall in [8], she determined completely for 1 ≦ n ≦ p + 1, where p is the smallest prime with the property that it divides the order of at least one of the free factors of G. However, she was unable to proceed beyond n = p + 1.

On the triple tensor product of prime-power groups

2020 ◽  
Vol 23 (5) ◽  
pp. 879-892
Author(s):  
S. Hadi Jafari ◽  
Halimeh Hadizadeh
Keyword(s):  
Tensor Product ◽  
Nilpotent Group ◽  
Upper Bound ◽  
Prime Power ◽  
Tensor Products ◽  
Lower Central Series ◽  
Central Series ◽  
Derived Subgroup

AbstractLet G be a finite p-group, and let {\otimes^{3}G} be its triple tensor product. In this paper, we obtain an upper bound for the order of {\otimes^{3}G}, which sharpens the bound given by G. Ellis and A. McDermott, [Tensor products of prime-power groups, J. Pure Appl. Algebra 132 1998, 2, 119–128]. In particular, when G has a derived subgroup of order at most p, we classify those groups G for which the bound is attained. Furthermore, by improvement of a result about the exponent of {\otimes^{3}G} determined by G. Ellis [On the relation between upper central quotients and lower central series of a group, Trans. Amer. Math. Soc. 353 2001, 10, 4219–4234], we show that, when G is a nilpotent group of class at most 4, {\exp(\otimes^{3}G)} divides {\exp(G)}.

scholarly journals A note on p-groups with power automorphisms

1992 ◽  
Vol 34 (3) ◽  
pp. 327-332 ◽  
Author(s):  
R. A. Bryce ◽  
John Cossey ◽  
E. A. Ormerod
Keyword(s):  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Subnormal Subgroup ◽  
Central Series ◽  
Related Concept ◽  
Finite Nilpotent Group ◽  
Basic Properties

Let G be a group. The norm, or Kern of G is the subgroup of elements of G which normalize every subgroup of the group. This idea was introduced in 1935 by Baer [1, 2], who delineated the basic properties of the norm. A related concept is the subgroup introduced by Wielandt [10] in 1958, and now named for him. The Wielandt subgroup of a group G is the subgroup of elements normalizing every subnormal subgroup of G. In the case of finite nilpotent groups these two concepts coincide, of course, since all subgroups of a finite nilpotent group are subnormal. Of late the Wielandt subgroup has been widely studied, and the name tends to be the more used, even in the finite nilpotent context when, perhaps, norm would be more natural. We denote the Wielandt subgroup of a group G by ω(G). The Wielandt series of subgroups ω1(G) is defined by: ω1(G) = ω(G) and for i ≥ 1, ωi+1(G)/ ω(G) = ω(G/ωi, (G)). The subgroups of the upper central series we denote by ζi(G).

scholarly journals Some automorphisms of finite nilpotent groups

1960 ◽  
Vol 4 (4) ◽  
pp. 204-207 ◽  
Author(s):  
J. C. Howarth
Keyword(s):  
Finite Groups ◽  
Lower Central Series ◽  
Nilpotent Groups ◽  
Central Series ◽  
Endomorphism Semigroup ◽  
Group Of Automorphisms ◽  
Inner Automorphism ◽  
Group A ◽  
Special Case

This note extends the concept of the inner automorphism, but here applies only to those finite groups G for which some member of the lower central series is Abelian. In general (e.g. when G is metabelian) the construction yields an endomorphism semigroup, but in the special case where Gis nilpotent (and may therefore, for our present purposes, be considered as a p-group) a group of automorphisms results.

On nilpotency of the group of outer class-preserving automorphisms of a group

2015 ◽  
Vol 15 (02) ◽  
pp. 1650026 ◽  
Author(s):  
Pradeep K. Rai
Keyword(s):  
Lower Central Series ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Central Series ◽  
Trivial Group ◽  
Improve Bound

Let G be a group and Zj(G), for j ≥ 0, be the jth term in the upper central series of G. We prove that if Out c(G/Zj(G)), the group of outer class-preserving automorphisms of G/Zj(G), is nilpotent of class k, then Out c(G) is nilpotent of class at most j + k. Moreover, if Out c(G/Zj(G)) is a trivial group, then Out c(G) is nilpotent of class at most j. As an application we prove that if γi(G)/γi(G) ∩ Zj(G) is cyclic then Out c(G) is nilpotent of class at most i + j, where γi(G), for i ≥ 1, denotes the ith term in the lower central series of G. This extends an earlier work of the author, where this assertion was proved for j = 0. We also improve bound on the nilpotency class of Out c(G) for some classes of nilpotent groups G.

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