scholarly journals ON NONNILPOTENT SUBSETS IN GENERAL LINEAR GROUPS

2011 ◽  
Vol 83 (3) ◽  
pp. 369-375
Author(s):  
AZIZOLLAH AZAD
Keyword(s):  
Linear Group ◽  
Simple Groups ◽  
General Linear ◽  
Linear Groups ◽  
Finite Simple Groups ◽  
Projective Special Linear Group ◽  
General Linear Groups ◽  
Nonnilpotent Subgroup ◽  
Group A ◽  
A Finite Group

AbstractLet G be a group. A subset X of G is said to be nonnilpotent if for any two distinct elements x and y in X, 〈x,y〉 is a nonnilpotent subgroup of G. If, for any other nonnilpotent subset X′ in G, ∣X∣≥∣X′ ∣, then X is said to be a maximal nonnilpotent subset and the cardinality of this subset is denoted by ω(𝒩G) . Using nilpotent nilpotentizers we find a lower bound for the cardinality of a maximal nonnilpotent subset of a finite group and apply this to the general linear group GL (n,q) . For all prime powers q we determine the cardinality of a maximal nonnilpotent subset of the projective special linear group PSL (2,q) , and we characterize all nonabelian finite simple groups G with ω(𝒩G)≤57 .

CHARACTERIZATION OF THE PROJECTIVE GENERAL LINEAR GROUPS PGL(2,q) BY THEIR ORDERS AND DEGREE PATTERNS

2009 ◽  
Vol 19 (07) ◽  
pp. 873-889 ◽  
Author(s):  
LIANGCAI ZHANG ◽  
XUEFENG LIU
Keyword(s):  
Simple Groups ◽  
General Linear ◽  
Linear Groups ◽  
Finite Simple Groups ◽  
Degree Pattern ◽  
General Linear Groups ◽  
A Finite Group ◽  
Projective General Linear Groups

Let G be a finite group and π(G) = {p1, p2,…,pk}. For p ∈ π(G), we put deg (p) := |{q ∈ π(G)|p ~ q}|, which is called the degree of p. We also define D(G) := ( deg (p1), deg (p2), …, deg (pk)), where p1 < p2 < ⋯ < pk, which is called the degree pattern of G. Using the classification of finite simple groups, we characterize the projective general linear group PGL(2,q)(q a prime power) by its order and degree pattern in the present paper.

scholarly journals MAXIMAL SUBSETS OF PAIRWISE NONCOMMUTING ELEMENTS OF THREE-DIMENSIONAL GENERAL LINEAR GROUPS

2009 ◽  
Vol 80 (1) ◽  
pp. 91-104 ◽  
Author(s):  
AZIZOLLAH AZAD ◽  
CHERYL E. PRAEGER
Keyword(s):  
Linear Group ◽  
Maximal Torus ◽  
General Linear Group ◽  
Three Dimensional ◽  
General Linear ◽  
Linear Groups ◽  
General Linear Groups ◽  
Maximal Subset ◽  
Group A

AbstractLet G be a group. A subset N of G is a set of pairwise noncommuting elements if xy⁄=yx for any two distinct elements x and y in N. If ∣N∣≥∣M∣ for any other set of pairwise noncommuting elements M in G, then N is said to be a maximal subset of pairwise noncommuting elements. In this paper we determine the cardinality of a maximal subset of pairwise noncommuting elements in a three-dimensional general linear group. Moreover, we show how to modify a given maximal subset of pairwise noncommuting elements into another maximal subset of pairwise noncommuting elements that contains a given ‘generating element’ from each maximal torus.

scholarly journals THE LOEWY SERIES OF THE STEINBERG-PIM OF FINITE GENERAL LINEAR GROUPS

2005 ◽  
Vol 92 (1) ◽  
pp. 62-98 ◽  
Author(s):  
BERND ACKERMANN
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
Indecomposable Module ◽  
Defect Group ◽  
General Linear ◽  
Linear Groups ◽  
General Linear Groups ◽  
Projective Indecomposable Module ◽  
Finite General Linear Group

In this paper we calculate the Loewy series of the projective indecomposable module of the unipotent block contained in the Gelfand–Graev module of the finite general linear group in the case of non-describing characteristic and Abelian defect group.

The ideal lattices of the complex group rings of finitary special and general linear groups over finite fields

1994 ◽  
Vol 116 (1) ◽  
pp. 7-25 ◽  
Author(s):  
B. Hartley ◽  
A. E. Zalesskii
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
Direct Limit ◽  
General Linear ◽  
Linear Groups ◽  
Group Rings ◽  
General Linear Groups ◽  
Ideal Lattices ◽  
Image Position ◽  
The Ideal

Letqbe a prime power, which will be fixed throughout the paper, letkbe a field, and letbe the field withqelements. LetGn(k)be the general linear groupGL(n, k), andSn(k)the special linear groupSL(n, k). The corresponding groups overwill be denoted simply byGnandSn. We may embedGn(k)inGn+1(k)via the mapForming the direct limit of the resulting system, we obtain thestable general linear groupG∞(k) overk.

Intersection graphs of general linear groups

2020 ◽  
pp. 2150039
Author(s):  
Mai Hoang Bien ◽  
Do Hoang Viet
Keyword(s):  
Finite Field ◽  
Linear Group ◽  
General Linear Group ◽  
Undirected Graph ◽  
Intersection Graph ◽  
General Linear ◽  
Linear Groups ◽  
Intersection Graphs ◽  
General Linear Groups ◽  
Vertex Set

Let [Formula: see text] be a field and [Formula: see text] the general linear group of degree [Formula: see text] over [Formula: see text]. The intersection graph [Formula: see text] of [Formula: see text] is a simple undirected graph whose vertex set includes all nontrivial proper subgroups of [Formula: see text]. Two vertices [Formula: see text] and [Formula: see text] of [Formula: see text] are adjacent if [Formula: see text] and [Formula: see text]. In this paper, we show that if [Formula: see text] is a finite field containing at least three elements, then the diameter [Formula: see text] is [Formula: see text] or [Formula: see text]. We also classify [Formula: see text] according to [Formula: see text]. In case [Formula: see text] is infinite, we prove that [Formula: see text] is one-ended of diameter 2 and its unique end is thick.

scholarly journals On isomorphisms of connected Cayley graphs, III

1998 ◽  
Vol 58 (1) ◽  
pp. 137-145 ◽  
Author(s):  
Cai Heng Li
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Simple Group ◽  
Prime Divisor ◽  
Cayley Graphs ◽  
Simple Groups ◽  
Linear Groups ◽  
Finite Simple Groups ◽  
Natural Question ◽  
A Finite Group

For a finite group G and a subset S of G which does not contain the identity of G, we use Cay(G, S) to denote the Cayley graph of G with respect to S. For a positive integer m, the group G is called a (connected) m-DCI-group if for any (connected) Cayley graphs Cay(G, S) and Cay(G, T) of out-valency at most m, Sσ = T for some σ ∈ Aut(G) whenever Cay(G, S) ≅ Cay(G, T). Let p(G) be the smallest prime divisor of |G|. It was previously shown that each finite group G is a connected m-DCI-group for m ≤ p(G) − 1 but this is not necessarily true for m = p(G). This leads to a natural question: which groups G are connected p(G)-DCI-groups? Here we conjecture that the answer of this question is positive for finite simple groups, that is, finite simple groups are all connected 2-DCI-groups. We verify this conjecture for the linear groups PSL(2, q). Then we prove that a nonabelian simple group G is a 2-DCI-group if and only if G = A5.

OD-Characterization of the Projective Special Linear Groups L2(q)

2012 ◽  
Vol 19 (03) ◽  
pp. 509-524 ◽  
Author(s):  
Liangcai Zhang ◽  
Wujie Shi
Keyword(s):  
Prime Graph ◽  
Simple Groups ◽  
Linear Groups ◽  
Finite Simple Groups ◽  
Projective Special Linear Group ◽  
Special Linear ◽  
Projective Special Linear Groups ◽  
Special Linear Groups

Let L2(q) be the projective special linear group, where q is a prime power. In the present paper, we prove that L2(q) is OD-characterizable by using the classification of finite simple groups. A new method is introduced in order to deal with the subtle changes of the prime graph of a group in the discussion of its OD-characterization. This not only generalizes a result of Moghaddamfar, Zokayi and Darafsheh, but also gives a positive answer to a conjecture put forward by Shi.

On the weakly monomial subgroups of finite simple groups

2019 ◽  
Vol 18 (02) ◽  
pp. 1950037
Author(s):  
Shuqin Dong ◽  
Hongfei Pan ◽  
Feng Tang
Keyword(s):  
Finite Group ◽  
Proper Subgroup ◽  
Maximal Subgroups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Group A ◽  
A Finite Group

Let [Formula: see text] be a finite group. A proper subgroup [Formula: see text] of [Formula: see text] is said to be weakly monomial if the order of [Formula: see text] satisfies [Formula: see text]. In this paper, we determine all the weakly monomial maximal subgroups of finite simple groups.

A Characterization of Some Finite Simple Groups Through Their Orders and Degree Patterns

2012 ◽  
Vol 19 (03) ◽  
pp. 473-482 ◽  
Author(s):  
M. Akbari ◽  
A. R. Moghaddamfar ◽  
S. Rahbariyan
Keyword(s):  
Finite Group ◽  
Simple Groups ◽  
Linear Groups ◽  
Finite Simple Groups ◽  
Degree Pattern ◽  
A Finite Group ◽  
Mersenne Prime

The degree pattern of a finite group G was introduced in [15] and denoted by D (G). A finite group M is said to be OD-characterizable if G ≅ M for every finite group G such that |G|=|M| and D (G)= D (M). In this article, we show that the linear groups Lp(2) and Lp+1(2) are OD-characterizable, where 2p-1 is a Mersenne prime. For example, the linear groups L2(2) ≅ S3, L3(2) ≅ L2(7), L4(2) ≅ A8, L5(2), L6(2), L7(2), L8(2), L13(2), L14(2), L17(2), L18(2), L19(2), L20(2), L31(2), L32(2), L61(2), L62(2), L89(2), L90(2), etc., are OD-characterizable. We also show that the simple groups L4(5), L4(7) and U4(7) are OD-characterizable.

A note on the subnormal structure of general linear groups

1990 ◽  
Vol 107 (2) ◽  
pp. 193-196 ◽  
Author(s):  
N. A. Vavilov
Keyword(s):  
Commutative Ring ◽  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Linear Groups ◽  
General Linear Groups ◽  
Elementary Subgroup ◽  
Subnormal Structure

The purpose of this note is to improve results of J. S. Wilson[12] and L. N. Vaserstein [10] concerning the subnormal structure of the general linear group G = GL (n, R) of degree n ≽ 3 over a commutative ring R. To do this we sharpen results of J. S. Wilson[12], A. Bak[1] and L. N. Vaserstein[10] on subgroups normalized by a relative elementary subgroup. It should be said also that (especially for the case n = 3) our proof is very much simpler than that of[12, 10]. To formulate our results let us recall some notation.

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