Subgroups dual to dimension subgroups

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.

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Augmentation quotients of some nonabelian finite groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100055675 ◽

1979 ◽

Vol 85 (2) ◽

pp. 261-270 ◽

Cited By ~ 7

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.

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The class of a nilpotent wreath product

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497270002548x ◽

1977 ◽

Vol 17 (1) ◽

pp. 53-89 ◽

Cited By ~ 10

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.

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Augmentation quotients of group rings and symmetric powers

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100055651 ◽

1979 ◽

Vol 85 (2) ◽

pp. 247-252 ◽

Cited By ~ 4

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.

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

Canadian Journal of Mathematics ◽

10.4153/cjm-1973-034-4 ◽

1973 ◽

Vol 25 (2) ◽

pp. 353-359 ◽

Cited By ~ 17

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.

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A TRANSFINITE FILTRATION OF SCHUR MULTIPLICATOR

International Journal of Algebra and Computation ◽

10.1142/s0218196705002785 ◽

2005 ◽

Vol 15 (05n06) ◽

pp. 1061-1073

Author(s):

ROMAN MIKHAILOV ◽

INDER BIR S. PASSI

Keyword(s):

Group Ring ◽

Lower Central Series ◽

Integral Group Ring ◽

Central Series ◽

Augmentation Ideal ◽

Integral Group ◽

Relationship Of ◽

The Relationship

We study certain subgroups of the Schur multiplicator of a group G. These subgroups are related to the identification of subgroups of G determind by ideals in its integral group ring ℤ[G]. Suitably defined transfinite powers of the augmentation ideal of ℤ[G] provide an increasing transfinite filtration of the Schur multiplicator of G. We investigate the relationship of this filtration with the transfinite lower central series of groups which are HZ-local in the sense of Bousfield.

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Wreath Products in Modular Group Rings

Bulletin of the London Mathematical Society ◽

10.1112/blms/23.5.443 ◽

1991 ◽

Vol 23 (5) ◽

pp. 443-444 ◽

Cited By ~ 1

Author(s):

Avinoam Mann

Keyword(s):

Modular Group ◽

Wreath Products ◽

Group Rings

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On Automorphisms of Complete Algebras And The Isomorphism Problem for Modular Group Rings

Canadian Journal of Mathematics ◽

10.4153/cjm-1990-021-9 ◽

1990 ◽

Vol 42 (3) ◽

pp. 383-394 ◽

Cited By ~ 1

Author(s):

Frank Röhl

Keyword(s):

Group Ring ◽

Modular Group ◽

Isomorphism Problem ◽

Nilpotency Class ◽

Affirmative Answer ◽

Integral Group Ring ◽

Group Rings ◽

Integral Group ◽

Integral Group Rings ◽

Analogous Question

In [5], Roggenkamp and Scott gave an affirmative answer to the isomorphism problem for integral group rings of finite p-groups G and H, i.e. to the question whether ZG ⥲ ZH implies G ⥲ H (in this case, G is said to be characterized by its integral group ring). Progress on the analogous question with Z replaced by the field Fp of p elements has been very little during the last couple of years; and the most far reaching result in this area in a certain sense - due to Passi and Sehgal, see [8] - may be compared to the integral case, where the group G is of nilpotency class 2.

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The normalizer property for integral group rings of holomorphs of finite nilpotent groups and the symmetric groups

Journal of Algebra and Its Applications ◽

10.1142/s0219498817500256 ◽

2017 ◽

Vol 16 (02) ◽

pp. 1750025 ◽

Cited By ~ 3

Author(s):

Jinke Hai ◽

Shengbo Ge ◽

Weiping He

Keyword(s):

Finite Group ◽

Symmetric Group ◽

Nilpotent Group ◽

Symmetric Groups ◽

Nilpotent Groups ◽

Group Rings ◽

Integral Group ◽

Integral Group Rings ◽

A Finite Group ◽

The Normalizer Property

Let [Formula: see text] be a finite group and let [Formula: see text] be the holomorph of [Formula: see text]. If [Formula: see text] is a finite nilpotent group or a symmetric group [Formula: see text] of degree [Formula: see text], then the normalizer property holds for [Formula: see text].

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Automorphisms of the integral group rings of some wreath products

Communications in Algebra ◽

10.1080/00927879108824152 ◽

1991 ◽

Vol 19 (2) ◽

pp. 519-534 ◽

Cited By ~ 5

Author(s):

Giambruno Antonio ◽

Angela Valenti ◽

Sudarshan K. Sehgal

Keyword(s):

Wreath Products ◽

Group Rings ◽

Integral Group ◽

Integral Group Rings

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Dimension quotients of metabelian Lie rings

International Journal of Algebra and Computation ◽

10.1142/s0218196717500114 ◽

2017 ◽

Vol 27 (02) ◽

pp. 251-258

Author(s):

Inder Bir S. Passi ◽

Thomas Sicking

Keyword(s):

Lower Central Series ◽

Universal Enveloping Algebra ◽

Central Series ◽

Augmentation Ideal ◽

Lie Ring ◽

Ring Of Integers ◽

Enveloping Algebra ◽

Lie Rings

For a Lie ring [Formula: see text] over the ring of integers, we compare its lower central series [Formula: see text] and its dimension series [Formula: see text] defined by setting [Formula: see text], where [Formula: see text] is the augmentation ideal of the universal enveloping algebra of [Formula: see text]. While [Formula: see text] for all [Formula: see text], the two series can differ. In this paper, it is proved that if [Formula: see text] is a metabelian Lie ring, then [Formula: see text], and [Formula: see text], for all [Formula: see text].

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