Free Differential Calculus. I: Derivation in the Free Group Ring

1953 ◽  
Vol 57 (3) ◽  
pp. 547 ◽  
Author(s):  
Ralph H. Fox
Keyword(s):  
Free Group ◽  
Group Ring ◽  
Differential Calculus ◽  
Calculus I

Massey Products and Lower Central Series of Free Groups

1987 ◽  
Vol 39 (2) ◽  
pp. 322-337 ◽  
Author(s):  
Roger Fenn ◽  
Denis Sjerve
Keyword(s):  
Free Group ◽  
Differential Calculus ◽  
Free Groups ◽  
Lower Central Series ◽  
Central Series ◽  
Massey Product ◽  
Massey Products ◽  
One Dimensional ◽  
Image Position

The purpose of this paper is to continue the investigation into the relationships amongst Massey products, lower central series of free groups and the free differential calculus (see [4], [9], [12]). In particular we set forth the notion of a universal Massey product ≪α1, …, αk≫, where the αi are one dimensional cohomology classes. This product is defined with zero indeterminacy, natural and multilinear in its variables.In order to state the results we need some notation. Throughout F will denote the free group on fixed generators x1, …, xn andwill denote the lower central series of F. If I = (i1, …, ik) is a sequence such that 1 ≦ i1, …, ik ≦ n then ∂1 is the iterated Fox derivative and , where is the augmentation. By convention we set ∂1 = identity if I is empty.

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.

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 The subgroup determined by a certain ideal in a free group ring

2016 ◽  
Vol 449 ◽  
pp. 400-407 ◽  
Author(s):  
Roman Mikhailov ◽  
Inder Bir S. Passi
Keyword(s):  
Free Group ◽  
Group Ring

Calculus I: The Derivative and Its Applications

2017 ◽  
Author(s):  
Patrick Clark
Keyword(s):  
Differential Calculus ◽  
Worked Examples ◽  
Ap Calculus ◽  
Calculus I ◽  
Independent Learner ◽  
Definition Of ◽  
User Friendly ◽  
Modular Format

Calculus I: The Derivative and Its Applications uniquely addresses all of the rules and applications of Differential Calculus necessary for the AP Calculus AB and BC courses. The material is presented in a modular format of 90 lessons that allows maximum flexibility for the student and the teacher. Lessons begin with the precalculus topics of functions and limits, discuss the definition of the derivative and all differentiation rules, and investigate applications of the derivative including curve sketching, optimization, and differentials. The lessons are designed to be rigorous enough for the serious student, yet user-friendly enough for the independent learner. All lessons include worked examples as well as exercises with solutions.

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