Augmentation quotients of group rings and symmetric powers

1979 ◽  
Vol 85 (2) ◽  
pp. 247-252 ◽  
Author(s):  
Robert Sandling ◽  
Ken-Ichi Tahara
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Additive Group ◽  
Lower Central Series ◽  
Symmetric Power ◽  
Central Series ◽  
Group Rings ◽  
Augmentation Ideal ◽  
Symmetric Powers ◽  
Image Position

Let G be a group with the lower central seriesLetwhere Σ runs over all non-negative integers a1, a2,…, an such that and is the aith symmetric power of the abelian group Gi/Gi+1 whereLet I (G) be the augmentation ideal of G in , the group ring of G over . Define the additive group Qn (G) = In (G) / In+1 (G) for any n ≥ 1. Then it is well known that Q1(G) ≅ W1(G) for any group G. Losey (4,5) proved that Q2(G) ≅ W2(G) for any finitely generated group G. Furthermore recently Tahara(12) proved that Q3(G) is a certain precisely defined quotient of W3(G) for any finite group G.

On the Structure of Q2(G) for Finitely Generated Groups

1973 ◽  
Vol 25 (2) ◽  
pp. 353-359 ◽  
Author(s):  
Gerald Losey
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Lower Central Series ◽  
Integral Group Ring ◽  
Symmetric Power ◽  
Central Series ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Finitely Generated ◽  
Finitely Generated Groups

Let G be a group, ZG its integral group ring and Δ = Δ(G) the augmentation ideal of ZG. Denote by Gi the ith term of the lower central series of G. Following Passi [3], we set . It is well-known that (see, for example [1]). In [3] Passi shows that if G is an abelian group then , the second symmetric power of G.

Augmentation quotients of some nonabelian finite groups

1979 ◽  
Vol 85 (2) ◽  
pp. 261-270 ◽  
Author(s):  
Gerald Losey ◽  
Nora Losey
Keyword(s):  
Group Structure ◽  
Lower Central Series ◽  
Central Series ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Nilpotent Residual ◽  
Periodic Behaviour ◽  
Augmentation Quotient ◽  
Image Position ◽  
A Finite Group

1. LetGbe a group,ZGits integral group ring and Δ = ΔGthe augmentation idealZGBy anaugmentation quotientofGwe mean any one of theZG-moduleswheren, r≥ 1. In recent years there has been a great deal of interest in determining the abelian group structure of the augmentation quotientsQn(G) =Qn,1(G) and(see (1, 2, 7, 8, 9, 12, 13, 14, 15)). Passi(8) has shown that in order to determineQn(G) andPn(G) for finiteGit is sufficient to assume thatGis ap-group. Passi(8, 9) and Singer(13, 14) have obtained information on the structure of these quotients for certain classes of abelianp-groups. However little seems to be known of a quantitative nature for nonabelian groups. In (2) Bachmann and Grünenfelder have proved the following qualitative result: ifGis a finite group then there exist natural numbersn0and π such thatQn(G) ≅Qn+π(G) for alln≥n0; ifGωis the nilpotent residual ofGandG/Gωhas classcthen π divides l.c.m. {1, 2, …,c}. There do not appear to be any examples in the literature of this periodic behaviour forc> 1. One of goals here is to present such examples. These examples will be from the class of finitep-groups in which the lower central series is anNp-series.

Subgroups dual to dimension subgroups

1972 ◽  
Vol 71 (1) ◽  
pp. 33-38 ◽  
Author(s):  
Robert Sandling
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Modular Group ◽  
Lower Central Series ◽  
Wreath Products ◽  
Central Series ◽  
Group Rings ◽  
Augmentation Ideal ◽  
Integral Group ◽  
New Criterion

Associated with, the powers of the augmentation ideal are the dimension subgroups. In the integral group ring case, they have long been conjectured to be the terms of the lower central series. This paper investigates the subgroups associated with the chain of ideals dual to the chain of powers of the augmentation ideal. The study is reduced to the case of the modular group rings of p-groups. The subgroups are calculated for Abelian p-groups, p odd. They appear in the upper central series of wreath products and provide a new criterion for the nilpotence of an arbitrary wreath product. The nilpotence class of wreath products is considered here as well; calculations and bounds are given; in particular, a new method of computing the class of the Sylow p-subgroups of the symmetric group arises.

scholarly journals On Central Series

1962 ◽  
Vol 13 (2) ◽  
pp. 175-178 ◽  
Author(s):  
I. D. Macdonald
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Lower Central Series ◽  
The Other ◽  
Central Series ◽  
Image Position ◽  
A Finite Group ◽  
Positive Integers

Letandbe, respectively, the upper and lower central series of a group G. Our purpose in this note is to extend known results and find some information as to which of the factors Zk/Zk−1 and Γk/Γk+1 may be infinite. Though our conclusions about the lower central series will be quite general we assume in the other case that the group is f.n., i.e. an extension of a finite group by a nilpotent group. The essential facts about f.n. groups are to be found in P. Hall's paper (4). We also refer to (4) for general notation; we reserve the letter k for positive integers.

scholarly journals The class of a nilpotent wreath product

1977 ◽  
Vol 17 (1) ◽  
pp. 53-89 ◽  
Author(s):  
David Shield
Keyword(s):  
Normal Subgroup ◽  
Wreath Product ◽  
Group Ring ◽  
Upper Bound ◽  
Lower Central Series ◽  
Nilpotency Class ◽  
Central Series ◽  
Augmentation Ideal ◽  
Cambridge Philos ◽  
Base Group

Let G be a group with a normal subgroup H whose index is a power of a prime p, and which is nilpotent with exponent a power of p. Gilbert Baumslag (Proc. Cambridge Philos. Soc. 55 (1959), 224–231) has shown that such a group is nilpotent; the main result of this paper is an upper bound on its nilpotency class in terms of parameters of H and G/H. It is shown that this bound is attained whenever G is a wreath product and H its base group.A descending central series, here called the cpp-series, is involved in these calculations more closely than is the lower central series, and the class of the wreath product in terms of this series is also found.Two tools used to obtain the main result, namely a useful basis for a finite p-group and a result about the augmentation ideal of the integer group ring of a finite p-group, may have some independent interest. The main result is applied to the construction of some two-generator groups of large nilpotency class with exponents 8, 9, and 25.

scholarly journals Integral Cayley Graphs over Abelian Groups

10.37236/353 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Walter Klotz ◽  
Torsten Sander
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Boolean Algebra ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Additive Group ◽  
Cyclic Groups ◽  
Vertex Set ◽  
A Finite Group ◽  
Gamma S

Let $\Gamma$ be a finite, additive group, $S \subseteq \Gamma, 0\notin S, -S=\{-s: s\in S\}=S$. The undirected Cayley graph Cay$(\Gamma,S)$ has vertex set $\Gamma$ and edge set $\{\{a,b\}: a,b\in \Gamma$, $a-b \in S\}$. A graph is called integral, if all of its eigenvalues are integers. For an abelian group $\Gamma$ we show that Cay$(\Gamma,S)$ is integral, if $S$ belongs to the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. The converse is proven for cyclic groups. A finite group $\Gamma$ is called Cayley integral, if every undirected Cayley graph over $\Gamma$ is integral. We determine all abelian Cayley integral groups.

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.

Natural maps of extension functors and a theorem of R. G. Swan

1961 ◽  
Vol 57 (3) ◽  
pp. 489-502 ◽  
Author(s):  
P. J. Hilton ◽  
D. Rees
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Group Ring ◽  
Projective Module ◽  
Additive Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Natural Maps ◽  
Image Position ◽  
A Finite Group

The present paper has been inspired by a theorem of Swan(5). The theorem can be described as follows. Let G be a finite group and let Γ be its integral group ring. We shall denote by Z an infinite cyclic additive group considered as a left Γ-module by defining gm = m for all g in G and m in Z. By a Tate resolution of Z is meant an exact sequencewhere Xn is a projective module for − ∞ < n < + ∞, and.

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.

Un principe d'ax-kochen-ershov pour des structures intermédiates entre groupes et corps valués

1999 ◽  
Vol 64 (3) ◽  
pp. 991-1027 ◽  
Author(s):  
Françoise Delon ◽  
Patrick Simonetta
Keyword(s):  
Abelian Group ◽  
Cross Section ◽  
Additive Group ◽  
Ordered Group ◽  
Unary Predicate ◽  
Valued Fields ◽  
Valued Field ◽  
Ordered Groups ◽  
Existentially Closed ◽  
Image Position

AbstractAn Ax-Kochen-Ershov principle for intermediate structures between valued groups and valued fields.We will consider structures that we call valued B-groups and which are of the form 〈G, B, *, υ〉 where– G is an abelian group,– B is an ordered group,– υ is a valuation denned on G taking its values in B,– * is an action of B on G satisfying: ∀x ϵ G ∀ b ∈ B υ(x * b) = ν(x) · b.The analysis of Kaplanski for valued fields can be adapted to our context and allows us to formulate an Ax-Kochen-Ershov principle for valued B-groups: we axiomatise those which are in some sense existentially closed and also obtain many of their model-theoretical properties. Let us mention some applications:1. Assume that υ(x) = υ(nx) for every integer n ≠ 0 and x ϵ G, B is solvable and acts on G in such a way that, for the induced action, Z[B] ∖ {0} embeds in the automorphism group of G. Then 〈G, B, *, υ〉 is decidable if and only if B is decidable as an ordered group.2. Given a field k and an ordered group B, we consider the generalised power series field k((B)) endowed with its canonical valuation. We consider also the following structure:where k((B))+ is the additive group of k((B)), S is a unary predicate interpreting {Tb ∣ b ϵB}, and ×↾k((B))×S is the multiplication restricted to k((B)) × S, structure which is a reduct of the valued field k((B)) with its canonical cross section. Then our result implies that if B is solvable and decidable as an ordered group, then M is decidable.3. A valued B–group has a residual group and our Ax-Kochen-Ershov principle remains valid in the context of expansions of residual group and value group. In particular, by adding a residual order we obtain new examples of solvable ordered groups having a decidable theory.

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