On Huppert's Conjecture for Alternating Groups of Low Degrees

2015 ◽  
Vol 22 (02) ◽  
pp. 293-308 ◽  
Author(s):  
Hung Ngoc Nguyen ◽  
Hung P. Tong-Viet ◽  
Thomas P. Wakefield
Keyword(s):  
Character Degrees ◽  
Simple Groups ◽  
Direct Factor ◽  
Sporadic Groups ◽  
Alternating Groups ◽  
Low Degrees ◽  
Groups Of Lie Type

Bertram Huppert conjectured in the late 1990s that the nonabelian simple groups are determined up to an abelian direct factor by the set of their character degrees. Although the conjecture has been established for various simple groups of Lie type and simple sporadic groups, it is expected to be difficult for alternating groups. In [5], Huppert verified the conjecture for the simple alternating groups An of degree up to 11. In this paper, we continue his work and verify the conjecture for the alternating groups of degrees 12 and 13.

Recognition of PSL(2,p) ×PSL(2,p) by its complex group algebra

2017 ◽  
Vol 16 (02) ◽  
pp. 1750036 ◽  
Author(s):  
Behrooz Khosravi ◽  
Zahra Momen ◽  
Behnam Khosravi ◽  
Bahman Khosravi
Keyword(s):  
Prime Number ◽  
Group Algebra ◽  
Character Degrees ◽  
Simple Groups ◽  
Group Algebras ◽  
Complex Group ◽  
Complex Group Algebra ◽  
Quasisimple Group ◽  
Natural Groups ◽  
Groups Of Lie Type

In [H. P. Tong-Viet, Simple classical groups of Lie type are determined by their character degrees, J. Algebra 357 (2012) 61–68] the following question arose: Question. Which groups can be uniquely determined by the structure of their complex group algebras? It is proved that every quasisimple group except covers of the alternating groups is uniquely determined up to isomorphism by the structure of [Formula: see text], the complex group algebra of [Formula: see text]. One of the next natural groups to be considered are the characteristically simple groups. In this paper, as the first step in this investigation we prove that if [Formula: see text] is an odd prime number, then [Formula: see text] is uniquely determined by the structure of its complex group algebra.

scholarly journals Elements with Square Roots in Finite Groups

2005 ◽  
Vol 12 (04) ◽  
pp. 677-690 ◽  
Author(s):  
M. S. Lucido ◽  
M. R. Pournaki
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Square Root ◽  
Alternating Groups ◽  
Square Roots ◽  
A Finite Group ◽  
Groups Of Lie Type

In this paper, we study the probability that a randomly chosen element in a finite group has a square root, in particular the simple groups of Lie type of rank 1, the sporadic finite simple groups and the alternating groups.

Recognizing Finite Groups Through Order and Degree Pattern

2008 ◽  
Vol 15 (03) ◽  
pp. 449-456 ◽  
Author(s):  
A. R. Moghaddamfar ◽  
A. R. Zokayi
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Symmetric Groups ◽  
Simple Groups ◽  
Alternating Groups ◽  
Degree Pattern ◽  
Sporadic Simple Groups ◽  
A Finite Group ◽  
Groups Of Lie Type

The degree pattern of a finite group G is introduced in [10] and it is proved that the following simple groups are uniquely determined by their degree patterns and orders: all sporadic simple groups, alternating groups Ap (p ≥ 5 is a twin prime) and some simple groups of Lie type. In this paper, we continue this investigation. In particular, we show that the automorphism groups of sporadic simple groups (except Aut (J2) and Aut (McL)), all simple C22-groups, the alternating groups Ap, Ap+1, Ap+2 and the symmetric groups Sp, Sp+1, where p is a prime, are also uniquely determined by their degree patterns and orders.

scholarly journals Equivariant character correspondences and inductive McKay condition for type A

2017 ◽  
Vol 2017 (728) ◽  
Author(s):  
Marc Cabanes ◽  
Britta Späth
Keyword(s):  
Finite Groups ◽  
Type A ◽  
Character Degrees ◽  
Simple Groups ◽  
Groups Of Lie Type

AbstractAs a step to establish the McKay conjecture on character degrees of finite groups, we verify the inductive McKay condition introduced by Isaacs–Malle–Navarro for simple groups of Lie type

scholarly journals Finite simple exceptional groups of Lie type in which all subgroups of odd index are pronormal

2020 ◽  
Vol 23 (6) ◽  
pp. 999-1016
Author(s):  
Anatoly S. Kondrat’ev ◽  
Natalia V. Maslova ◽  
Danila O. Revin
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups ◽  
Exceptional Groups ◽  
Odd Index ◽  
Groups Of Lie Type

AbstractA subgroup H of a group G is said to be pronormal in G if H and {H^{g}} are conjugate in {\langle H,H^{g}\rangle} for every {g\in G}. In this paper, we determine the finite simple groups of type {E_{6}(q)} and {{}^{2}E_{6}(q)} in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal.

scholarly journals On the Structure of Integral Group Rings of Sporadic Groups

2000 ◽  
Vol 3 ◽  
pp. 274-306 ◽  
Author(s):  
Frauke M. Bleher ◽  
Wolfgang Kimmerle
Keyword(s):  
Automorphism Groups ◽  
Symmetric Groups ◽  
Computational Procedure ◽  
Isomorphism Problem ◽  
Simple Groups ◽  
Group Rings ◽  
Integral Group ◽  
Sporadic Groups ◽  
Integral Group Rings ◽  
Sporadic Simple Groups

AbstractThe object of this article is to examine a conjecture of Zassenhaus and certain variations of it for integral group rings of sporadic groups. We prove the ℚ-variation and the Sylow variation for all sporadic groups and their automorphism groups. The Zassenhaus conjecture is established for eighteen of the sporadic simple groups, and for all automorphism groups of sporadic simple groups G which are different from G. The proofs are given with the aid of the GAP computer algebra program by applying a computational procedure to the ordinary and modular character tables of the groups. It is also shown that the isomorphism problem of integral group rings has a positive answer for certain almost simple groups, in particular for the double covers of the symmetric groups.

scholarly journals Walks on graphs as symmetric or asymmetric tools to encrypt data

2002 ◽  
Vol 20 (1) ◽  
pp. 34 ◽  
Author(s):  
V. A. Ustimenko ◽  
Y. M. Khmelevsky
Keyword(s):  
Real Life ◽  
Theoretical Method ◽  
General Idea ◽  
Simple Groups ◽  
Prototype Model ◽  
Encryption Scheme ◽  
Symmetric Encryption ◽  
Encrypt Data ◽  
Rank 2 ◽  
Groups Of Lie Type

New results on graph theoretical method of encryption will be presented. The general idea is to treat vertices of a graph as messages, and walks of a certain length as ecnryption tools. We will construct one-time pad algorithms with a certain resistance to attacks when the adversary knows plaintext and ciphertext. Special linguistic graphs of high girth whose vertices (messages) and walks (encoding tools) could be both naturally identified with vectors over the finite field, and with the so-called parallelotopic graphs, which turn out to be efficient tools for symmetric encryption. We will formulate criteria when parallelotopic graph (or the more general graph of tactical configuration) is a graph of absolutely optimal encryption scheme, producing asymptotic one-time pad algorithm. We will show how to convert one-time pads, which are related to geometries of rank 2 of simple groups of Lie type, to a real-life encryption scheme involving potentially infinite text and flexible passwords. We will discuss families of linguistic and parallelotopic graphs of increasing girth as the source for the generation of asymmetric cryptographic functions and related open key algorithms. We will construct new families of such graphs via group theoretical and geometrical technique. The software for symmetric and asymmetric ecnryption (prototype model of the package) is ready for demonstration.

scholarly journals On the Steinberg Character of a Finite Simple Group of Lie Type

1971 ◽  
Vol 12 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Bhama Srinivasan
Keyword(s):  
Algebraic Group ◽  
Linear Algebraic Group ◽  
Simple Groups ◽  
Closed Field ◽  
Finite Simple Groups ◽  
Reductive Algebraic Group ◽  
Simple Algebraic Group ◽  
Simply Connected ◽  
Groups Of Lie Type ◽  
Steinberg Character

Let K be an algebraically closed field of characteristic ρ >0. If G is a connected, simple connected, semisimple linear algebraic group defined over K and σ an endomorphism of G onto G such that the subgroup Gσ of fixed points of σ is finite, Steinberg ([6] [7]) has shown that there is a complex irreducible character χ of Gσ with the following properties. χ vanishes at all elements of Gσ which are not semi- simple, and, if x ∈ G is semisimple, χ(x) = ±n(x) where n(x)is the order of a Sylow p-subgroup of (ZG(x))σ (ZG(x) is the centraliser of x in G). If G is simple he has, in [6], identified the possible groups Gσ they are the Chevalley groups and their twisted analogues over finite fields, that is, the ‘simply connected’ versions of finite simple groups of Lie type. In this paper we show, under certain restrictions on the type of the simple algebraic group G an on the characteristic of K, that χ can be expressed as a linear combination with integral coefficients of characters induced from linear characters of certain naturally defined subgroups of Gσ. This expression for χ gives an explanation for the occurence of n(x) in the formula for χ (x), and also gives an interpretation for the ± 1 occuring in the formula in terms of invariants of the reductive algebraic group ZG(x).

scholarly journals An identification theorem for the sporadic simple groups F2 and M(23)

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Chris Parker ◽  
Gernot Stroth
Keyword(s):  
Simple Groups ◽  
Sporadic Groups ◽  
Sporadic Simple Groups ◽  
Identification Theorem

Abstract.We identify the sporadic groups M(23) and F

Sign in / Sign up

Export Citation Format

Share Document