scholarly journals On automorphisms of graded quasi-lie algebras

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 3141-3150
Author(s):  
Dae-Woong Lee ◽  
Sunyoung Lee ◽  
Yeonjeong Kim ◽  
Jeong-Eun Lim
Keyword(s):  
Finite Group ◽  
Lie Algebras ◽  
Infinite Group ◽  
Ring Of Integers ◽  
A Finite Group

Let Z be the ring of integers and let K(Z,2n) denote the Eilenberg-MacLane space of type (Z,2n) for n ? 1. In this article, we prove that the graded group Am := Aut(??2mn+1(?K(Z,2n))=torsions) of automorphisms of the graded quasi-Lie algebras ?? 2mn+1(?K(Z,2n)) modulo torsions that preserve the Whitehead products is a finite group for m ? 2 and an infinite group for m ? 3, and that the group Aut(?*(K(Z,2n))=torsions) is non-abelian. We extend and apply those results to techniques in localization (or rationalization) theory.

Finite group schemes of p-rank ≤ 1

2017 ◽  
Vol 166 (2) ◽  
pp. 297-323
Author(s):  
HAO CHANG ◽  
ROLF FARNSTEINER
Keyword(s):  
Finite Group ◽  
Lie Algebras ◽  
Group Scheme ◽  
Closed Field ◽  
Difficult Case ◽  
Modular Representation Theory ◽  
Group Schemes ◽  
Restricted Lie Algebras ◽  
A Finite Group ◽  
Finite Group Schemes

AbstractLet be a finite group scheme over an algebraically closed field k of characteristic char(k) = p ≥ 3. In generalisation of the familiar notion from the modular representation theory of finite groups, we define the p-rank rkp() of and determine the structure of those group schemes of p-rank 1, whose linearly reductive radical is trivial. The most difficult case concerns infinitesimal groups of height 1, which correspond to restricted Lie algebras. Our results show that group schemes of p-rank ≤ 1 are closely related to those being of finite or domestic representation type.

ON THETA PAIRS FOR A MAXIMAL SUBALGEBRA

2012 ◽  
Vol 11 (01) ◽  
pp. 1250001 ◽  
Author(s):  
ALI REZA SALEMKAR ◽  
SARA CHEHRAZI ◽  
SOMAIEH ALIZADEH NIRI
Keyword(s):  
Finite Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Nonzero Ideal ◽  
Maximal Subgroups ◽  
Maximal Subalgebra ◽  
Nilpotent Lie Algebras ◽  
Finite Dimensional ◽  
A Finite Group ◽  
Number Of Authors

Given a maximal subalgebra M of a finite-dimensional Lie algebra L, a θ-pair for M is a pair (A, B) of subalgebras such that A ≰ M, B is an ideal of L contained in A ∩ M, and A/B includes properly no nonzero ideal of L/B. This is analogous to the concept of θ-pairs associated to maximal subgroups of a finite group, which has been studied by a number of authors. A θ-pair (A, B) for M is said to be maximal if M has no θ-pair (C, D) such that A < C. In this paper, we obtain some properties of maximal θ-pairs and use them to give some characterizations of solvable, supersolvable and nilpotent Lie algebras.

scholarly journals Growth in Infinite Groups of Infinite Subsets

2015 ◽  
Vol 22 (02) ◽  
pp. 333-348
Author(s):  
J. O. Button
Keyword(s):  
Finite Group ◽  
Invariant Measure ◽  
Finite Index ◽  
Infinite Group ◽  
Infinite Groups ◽  
A Finite Group ◽  
Finite Index Subgroups ◽  
Free Sets

Given an infinite group G, we consider the finitely additive invariant measure defined on finite unions of cosets of finite index subgroups. We show that this shares many properties with the size of subsets of a finite group, for instance we can obtain equivalent results on the Ruzsa distance and product free sets. In particular, if G has infinitely many finite index subgroups, then it has subsets S of measure arbitrarily close to 1/2 with square S2 having measure less than 1. We also examine properties of the Ruzsa distance on the set of finite index subgroups of an infinite group, whereupon it becomes a genuine metric.

scholarly journals The CI Problem for Infinite Groups

10.37236/5056 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Joy Morris
Keyword(s):  
Finite Group ◽  
Cayley Graphs ◽  
Torsion Group ◽  
Infinite Group ◽  
Infinite Groups ◽  
Open Problems ◽  
Locally Finite ◽  
A Finite Group

A finite group $G$ is a DCI-group if, whenever $S$ and $S'$ are subsets of $G$ with the Cayley graphs Cay$(G,S)$ and Cay$(G,S')$ isomorphic, there exists an automorphism $\varphi$ of $G$ with $\varphi(S)=S'$. It is a CI-group if this condition holds under the restricted assumption that $S=S^{-1}$. We extend these definitions to infinite groups, and make two closely-related definitions: an infinite group is a strongly (D)CI$_f$-group if the same condition holds under the restricted assumption that $S$ is finite; and an infinite group is a (D)CI$_f$-group if the same condition holds whenever $S$ is both finite and generates $G$.We prove that an infinite (D)CI-group must be a torsion group that is not locally-finite. We find infinite families of groups that are (D)CI$_f$-groups but not strongly (D)CI$_f$-groups, and that are strongly (D)CI$_f$-groups but not (D)CI-groups. We discuss which of these properties are inherited by subgroups. Finally, we completely characterise the locally-finite DCI-graphs on $\mathbb Z^n$. We suggest several open problems related to these ideas, including the question of whether or not any infinite (D)CI-group exists.

scholarly journals On the units of a modular group ring

1972 ◽  
Vol 7 (2) ◽  
pp. 169-182 ◽  
Author(s):  
K.R. Pearson
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Symmetric Group ◽  
Group Ring ◽  
Modular Group ◽  
Ring Of Integers ◽  
Group Of Units ◽  
A Finite Group

It is shown that a finite group G is a normal subgroup of the group of units of the group ring of G over the ring of integers modulo n if and only if G is abelian or n = 2 and G is isomorphic to the symmetric group on 3 letters.

scholarly journals Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes

2018 ◽  
Vol 30 (2) ◽  
pp. 479-495
Author(s):  
Yang Pan
Keyword(s):  
Finite Group ◽  
Lie Algebras ◽  
Finite Groups ◽  
Positive Characteristic ◽  
Group Scheme ◽  
Closed Field ◽  
Group Schemes ◽  
Frobenius Kernel ◽  
A Finite Group ◽  
Finite Group Schemes

AbstractWe investigate the saturation rank of a finite group scheme defined over an algebraically closed field{\Bbbk}of positive characteristicp. We begin by exploring the saturation rank for finite groups and infinitesimal group schemes. Special attention is given to reductive Lie algebras and the second Frobenius kernel of the algebraic group{\operatorname{SL}_{n}}.

scholarly journals On normally embedded subgroups of locally soluble FC-groups

1986 ◽  
Vol 28 (2) ◽  
pp. 153-159 ◽  
Author(s):  
J. C. Beidleman ◽  
M. J. Karbe
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Normal Subgroup ◽  
Sylow Subgroup ◽  
Infinite Group ◽  
Considerable Importance ◽  
Soluble Groups ◽  
Sylow Subgroups ◽  
Group A ◽  
A Finite Group

In his Habilitationsschrift [3] B. Fischer introduced the concept of a normally embedded subgroup of a finite group. A subgroup of a finite group G is said to be normally embedded in G if each of its Sylow subgroups is a Sylow subgroup of a normal subgroup of G. Meanwhile this concept has become of considerable importance in the theory of finite soluble groups and has been studied by various authors. However, in infinite group theory, normally embedded subgroups seem to have received little attention. The object of this note is to study normally embedded subgroups of locally soluble FC-groups.

ON HIGHER FROBENIUS–SCHUR INDICATORS

2021 ◽  
pp. 1-11
Author(s):  
YANJUN LIU ◽  
WOLFGANG WILLEMS
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Irreducible Characters ◽  
A Finite Group

Abstract Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators $\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$ for primes p and $n \in \mathbb {N}$ , where G is a finite group and $\chi $ is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.

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