scholarly journals A sufficient condition for the second derived factor group to be finite

1987 ◽  
Vol 30 (1) ◽  
pp. 83-89
Author(s):  
J. R. Howse
Keyword(s):  
Free Group ◽  
Group Ring ◽  
Factor Group ◽  
Sufficient Condition

This paper concerns an application of an algorithm for the second derived factor group as described by Howse and Johnson in [3]. This algorithm has as its basis theFox derivative (see [1]), a mapping from the free group F to the group-ring ℤF, definedas follows: let X be a set of generators of a group G, and let w = y1…yk with each yi∈X±1.

scholarly journals GROUP RINGS OF COUNTABLE NON-ABELIAN LOCALLY FREE GROUPS ARE PRIMITIVE

2011 ◽  
Vol 21 (03) ◽  
pp. 409-431 ◽  
Author(s):  
TSUNEKAZU NISHINAKA
Keyword(s):  
Free Group ◽  
Group Ring ◽  
Free Groups ◽  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Hnn Extensions ◽  
Necessary And Sufficient ◽  
Countably Infinite

We prove that every group ring of a non-abelian locally free group which is the union of an ascending sequence of free groups is primitive. In particular, every group ring of a countable non-abelian locally free group is primitive. In addition, by making use of the result, we give a necessary and sufficient condition for group rings of ascending HNN extensions of free groups to be primitive, which extends the main result in [Group rings of proper ascending HNN extensions of countably infinite free groups are primitive, J. Algebra317 (2007) 581–592] to the general cardinality case.

scholarly journals The SQ-universality of some finitely presented groups

1973 ◽  
Vol 16 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Peter M. Neumann
Keyword(s):  
Free Group ◽  
Finite Index ◽  
Countable Group ◽  
Factor Group ◽  
Finitely Presented Groups ◽  
Rank 2 ◽  
Finitely Presented

Following a suggestion of G. Higman we say that the group G is SQ-universal if every countable group is embeddable in some factor group of G. It is a well-known theorem of G. Higman, B. H. Neumann and Hanna Neumann that the free group of rank 2 is sq-universal in this sense. Several different proofs are now available (see, for example, [1] or [9]). It is my intention to prove the LEmma. If H is a subgroup of finite index in a group G, then G is SQ-universal if and only if H is SQ-universal.

scholarly journals GENERICITY OF FILLING ELEMENTS

2012 ◽  
Vol 22 (02) ◽  
pp. 1250008 ◽  
Author(s):  
BRENT B. SOLIE
Keyword(s):  
Free Group ◽  
Time Complexity ◽  
Linear Time ◽  
Sufficient Condition ◽  
Membership Problem ◽  
Isometric Action ◽  
Finitely Generated ◽  
Generic Subset ◽  
Translation Length

An element of a finitely generated non-Abelian free group F(X) is said to be filling if that element has positive translation length in every very small minimal isometric action of F(X) on an ℝ-tree. We give a proof that the set of filling elements of F(X) is exponentially F(X)-generic in the sense of Arzhantseva and Ol'shanskiı. We also provide an algebraic sufficient condition for an element to be filling and show that there exists an exponentially F(X)-generic subset consisting only of filling elements and whose membership problem has linear time complexity.

scholarly journals Commutative feebly nil-clean group rings

2019 ◽  
Vol 11 (2) ◽  
pp. 264-270
Author(s):  
Peter V. Danchev
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Clean Rings ◽  
Unital Ring ◽  
Necessary And Sufficient

Abstract An arbitrary unital ring R is called feebly nil-clean if any its element is of the form q + e − f, where q is a nilpotent and e, f are idempotents with ef = fe. For any commutative ring R and any abelian group G, we find a necessary and sufficient condition when the group ring R(G) is feebly nil-clean only in terms of R, G and their sections. Our result refines establishments due to McGovern et al. in J. Algebra Appl. (2015) on nil-clean rings and Danchev-McGovern in J. Algebra (2015) on weakly nil-clean rings, respectively.

Polynomial Ideals in Group Rings

1973 ◽  
Vol 25 (6) ◽  
pp. 1174-1182 ◽  
Author(s):  
M. M. Parmenter ◽  
I. B. S. Passi ◽  
S. K. Sehgal
Keyword(s):  
Commutative Ring ◽  
Free Group ◽  
Group Ring ◽  
Multiplicative Group ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Polynomial Ideals

Letf(x1, x2, … , xn) be a polynomial in n non-commuting variables x1, x2, … , xn and their inverses with coefficients in the ring Z of integers, i.e. an element of the integral group ring of the free group on X1, x2, … , xn. Let R be a commutative ring with unity, G a multiplicative group and R(G) the group ring of G with coefficients in R.

BASS CYCLIC UNITS AS FACTORS IN A FREE GROUP IN INTEGRAL GROUP RING UNITS

2011 ◽  
Vol 21 (04) ◽  
pp. 531-545 ◽  
Author(s):  
JAIRO Z. GONÇALVES ◽  
ÁNGEL DEL RÍO
Keyword(s):  
Finite Group ◽  
Free Group ◽  
Group Ring ◽  
Prime Order ◽  
Infinite Order ◽  
Integral Group Ring ◽  
Integral Group ◽  
Cyclic Unit ◽  
Central Elements ◽  
Group A

Marciniak and Sehgal showed that if u is a non-trivial bicyclic unit of an integral group ring then there is a bicyclic unit v such that u and v generate a non-abelian free group. A similar result does not hold for Bass cyclic units of infinite order based on non-central elements as some of them have finite order modulo the center. We prove a theorem that suggests that this is the only limitation to obtain a non-abelian free group from a given Bass cyclic unit. More precisely, we prove that if u is a Bass cyclic unit of an integral group ring ℤG of a solvable and finite group G, such that u has infinite order modulo the center of U(ℤG) and it is based on an element of prime order, then there is a non-abelian free group generated by a power of u and a power of a unit in ℤG which is either a Bass cyclic unit or a bicyclic unit.

The Cohn localization of the free group ring

1992 ◽  
Vol 111 (3) ◽  
pp. 433-443 ◽  
Author(s):  
M. Farber ◽  
P. Vogel
Keyword(s):  
Free Group ◽  
Group Ring

In [1] P. Cohn suggested the construction of a localization of a ring with respect to a class of square matrices. Let us briefly recall the definitions.

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