On Nilpotent Products of Cyclic Groups. II

1961 ◽  
Vol 13 ◽  
pp. 557-568 ◽  
Author(s):  
Ruth Rebekka Struik
Keyword(s):  
Normal Subgroup ◽  
Finite Number ◽  
Free Product ◽  
Multiplication Table ◽  
Unique Representation ◽  
Cyclic Groups ◽  
Positive Integers

In a previous paper (18), G = F/Fn was studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. In that paper the following cases were completely treated:(a) F a free product of cyclic groups of order pαi, p a prime, αi positive integers, and n = 4, 5, … , p + 1.(b) F a free product of cyclic groups of order 2αi, and n = 4.In this paper, the following case is completely treated:(c) F a free product of cyclic groups of order pαi p a prime, αi positive integers, and n = p + 2.(Note that n = 2 is well known, and n — 3 was studied by Golovin (2).) By ‘'completely treated” is meant: a unique representation of elements of the group is given, and the order of the group is indicated. In the case of n = 4, a multiplication table was given.

scholarly journals Relative relation modules of finite groups

1991 ◽  
Vol 34 (3) ◽  
pp. 433-442 ◽  
Author(s):  
Mohammad Yamin
Keyword(s):  
Normal Subgroup ◽  
Finite Number ◽  
Free Product ◽  
Free Group ◽  
Finite Groups ◽  
Relation Module ◽  
Cyclic Groups

Let E be a free product of a finite number of cyclic groups, and S a normal subgroup of E such that E/S ≅ G is finite. For a prime p, Ŝ = S / S′Sp may be regarded as an -module. Whenever E is a free group, Ŝ is called a relation module (modulo p); in general we call Ŝ a relative relation module (modulo p). Gaschütz, Gruenberg and others have studied relation modules; the aim of this paper is to study relative relation modules.

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.

scholarly journals Serre’s Property (FA) for automorphism groups of free products

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Naomi Andrew
Keyword(s):  
Automorphism Group ◽  
Free Product ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Free Products ◽  
Cyclic Groups ◽  
Link Type ◽  
Open Case

AbstractWe provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore, this paper generalises the work of N. Leder [Serre’s Property FA for automorphism groups of free products, preprint (2018), https://arxiv.org/abs/1810.06287v1]. for finite cyclic groups, as well as resolving the open case of that paper.

Embedding Theorems for Countable Groups

1970 ◽  
Vol 22 (4) ◽  
pp. 827-835 ◽  
Author(s):  
James McCool
Keyword(s):  
Free Product ◽  
Word Problem ◽  
Countable Group ◽  
Factor Group ◽  
Decision Problems ◽  
Alternative Proof ◽  
Embedding Theorems ◽  
Cyclic Groups ◽  
Result Theorem ◽  
The Given

A group P is said to be a CEF-group if, for every countable group G, there is a factor group of P which contains a subgroup isomorphic to G. It was shown by Higman, Neumann, and Neumann [5] that the free group of rank two is a CEF-group. More recently, Levin [6] proved that if P is the free product of two cyclic groups, not both of order two, then P is a CEF-group. Later, Hall [3] gave an alternative proof of Levin's result.In this paper we give a new proof of Levin's result (Theorem 2). The proof given yields information about the factor group H of P in which a given countable group G is embedded; for example, if G is given by a recursive presentation (this concept is denned in [4]), then a recursive presentation is obtained for H, and certain decision problems (in particular, the word problem) are solvable for the recursive presentation obtained for H if and only if they are solvable for the given recursive presentation of G.

On Nilpotent Products of Cyclic Groups

1960 ◽  
Vol 12 ◽  
pp. 447-462 ◽  
Author(s):  
Ruth Rebekka Struik
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Abelian Group ◽  
Nilpotent Groups ◽  
Multiplication Table ◽  
Direct Products ◽  
Free Products ◽  
Cyclic Groups ◽  
Special Cases ◽  
Product Of Groups

In this paper G = F/Fn is studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. The case of n = 4 is completely treated (F/F2 is well known; F/F3 is completely treated in (2)); special cases of n > 4 are studied; a partial conjecture is offered in regard to the unsolved cases. For n = 4 a multiplication table and other properties are given.The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.

ON FINITE BRANCHED UNIFORMIZATIONS OF THE PROJECTIVE PLANE

2013 ◽  
Vol 24 (02) ◽  
pp. 1350017
Author(s):  
A. MUHAMMED ULUDAĞ ◽  
CELAL CEM SARIOĞLU
Keyword(s):  
Projective Plane ◽  
Fundamental Group ◽  
Free Product ◽  
Plane Curve ◽  
Complex Projective Plane ◽  
Galois Covering ◽  
Cyclic Groups ◽  
Galois Coverings ◽  
Complex Projective ◽  
Branching Index

We give a brief survey of the so-called Fenchel's problem for the projective plane, that is the problem of existence of finite Galois coverings of the complex projective plane branched along a given divisor and prove the following result: Let p, q be two integers greater than 1 and C be an irreducible plane curve. If there is a surjection of the fundamental group of the complement of C into a free product of cyclic groups of orders p and q, then there is a finite Galois covering of the projective plane branched along C with any given branching index.

scholarly journals Harmonic analysis on the free product of two cyclic groups

1986 ◽  
Vol 65 (2) ◽  
pp. 147-171 ◽  
Author(s):  
Donald I Cartwright ◽  
P.M Soardi
Keyword(s):  
Harmonic Analysis ◽  
Free Product ◽  
Cyclic Groups

CODING PARTITIONS OF REGULAR SETS

2009 ◽  
Vol 19 (08) ◽  
pp. 1011-1023 ◽  
Author(s):  
MARIE-PIERRE BÉAL ◽  
FABIO BURDERI ◽  
ANTONIO RESTIVO
Keyword(s):  
Finite Number ◽  
Free Product ◽  
Canonical Decomposition ◽  
Minimal Length ◽  
Regular Monoid ◽  
Finite Sequences ◽  
Regular Sets ◽  
Regular Classes

A coding partition of a set of words partitions this set into classes such that whenever a sequence, of minimal length, has two distinct factorizations, the words of these factorizations belong to the same class. The canonical coding partition is the finest coding partition that partitions the set of words in at most one unambiguous class and other classes that localize the ambiguities in the factorizations of finite sequences. We prove that the canonical coding partition of a regular set contains a finite number of regular classes and we give an algorithm for computing this partition. From this we derive a canonical decomposition of a regular monoid into a free product of finitely many regular monoids.

scholarly journals A finiteness theorem on symplectic singularities

2016 ◽  
Vol 152 (6) ◽  
pp. 1225-1236 ◽  
Author(s):  
Yoshinori Namikawa
Keyword(s):  
Finite Number ◽  
Contact Structure ◽  
Finiteness Theorem ◽  
Maximal Weight ◽  
Fixed Dimension ◽  
Positive Integers ◽  
Boundedness Result ◽  
Symplectic Varieties

An affine symplectic singularity$X$with a good$\mathbf{C}^{\ast }$-action is called a conical symplectic variety. In this paper we prove the following theorem. For fixed positive integers$N$and$d$, there are only a finite number of conical symplectic varieties of dimension$2d$with maximal weights$N$, up to an isomorphism. To prove the main theorem, we first relate a conical symplectic variety with a log Fano Kawamata log terminal (klt) pair, which has a contact structure. By the boundedness result for log Fano klt pairs with fixed Cartier index, we prove that conical symplectic varieties of a fixed dimension and with a fixed maximal weight form a bounded family. Next we prove the rigidity of conical symplectic varieties by using Poisson deformations.

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