scholarly journals A REFINED WARING PROBLEM FOR FINITE SIMPLE GROUPS

2015 ◽  
Vol 3 ◽  
Author(s):  
MICHAEL LARSEN ◽  
PHAM HUU TIEP
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Lie Group ◽  
Free Groups ◽  
Duke Math ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Compact Lie Group ◽  
Waring Problem ◽  
Nonabelian Simple Group

Let $w_{1}$ and $w_{2}$ be nontrivial words in free groups $F_{n_{1}}$ and $F_{n_{2}}$, respectively. We prove that, for all sufficiently large finite nonabelian simple groups $G$, there exist subsets $C_{1}\subseteq w_{1}(G)$ and $C_{2}\subseteq w_{2}(G)$ such that $|C_{i}|=O(|G|^{1/2}\log ^{1/2}|G|)$ and $C_{1}C_{2}=G$. In particular, if $w$ is any nontrivial word and $G$ is a sufficiently large finite nonabelian simple group, then $w(G)$ contains a thin base of order $2$. This is a nonabelian analog of a result of Van Vu [‘On a refinement of Waring’s problem’, Duke Math. J. 105(1) (2000), 107–134.] for the classical Waring problem. Further results concerning thin bases of $G$ of order $2$ are established for any finite group and for any compact Lie group $G$.

Mersenne primes and solvable Sylow numbers

2016 ◽  
Vol 15 (09) ◽  
pp. 1650163
Author(s):  
Tian-Ze Li ◽  
Yan-Jun Liu
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Solvable Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Nonabelian Simple Group ◽  
Mersenne Primes ◽  
A Finite Group ◽  
Mersenne Prime

Let [Formula: see text] be a prime. The Sylow [Formula: see text]-number of a finite group [Formula: see text], which is the number of Sylow [Formula: see text]-subgroups of [Formula: see text], is called solvable if its [Formula: see text]-part is congruent to [Formula: see text] modulo [Formula: see text] for any prime [Formula: see text]. P. Hall showed that solvable groups only have solvable Sylow numbers, and M. Hall showed that the Sylow [Formula: see text]-number of a finite group is the product of two kinds of factors: of prime powers [Formula: see text] with [Formula: see text] (mod [Formula: see text]) and of the number of Sylow [Formula: see text]-subgroups in certain finite simple groups (involved in [Formula: see text]). These classical results lead to the investigation of solvable Sylow numbers of finite simple groups. In this paper, we show that a finite nonabelian simple group has only solvable Sylow numbers if and only if it is isomorphic to [Formula: see text] for [Formula: see text] a Mersenne prime.

scholarly journals A Study About One Generation of Finite Simple Groups and Finite Groups

2021 ◽  
Vol 13 (3) ◽  
pp. 59
Author(s):  
Nader Taffach
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Prime Numbers ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
A Finite Group

In this paper, we study the problem of how a finite group can be generated by some subgroups. In order to the finite simple groups, we show that any finite non-abelian simple group can be generated by two Sylow p1 - and p_2 -subgroups, where p_1  and p_2  are two different primes. We also show that for a given different prime numbers p  and q , any finite group can be generated by a Sylow p -subgroup and a q -subgroup.

On mutually permutable products of generalized supersoluble subgroups of finite groups

2016 ◽  
Vol 09 (03) ◽  
pp. 1650054
Author(s):  
E. N. Myslovets
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Paper Properties ◽  
Supersoluble Groups ◽  
A Finite Group ◽  
Mutually Permutable Products ◽  
Composition Factors

Let [Formula: see text] be a class of finite simple groups. We say that a finite group [Formula: see text] is a [Formula: see text]-group if all composition factors of [Formula: see text] are contained in [Formula: see text]. A group [Formula: see text] is called [Formula: see text]-supersoluble if every chief [Formula: see text]-factor of [Formula: see text] is a simple group. In this paper, properties of mutually permutable products of [Formula: see text]-supersoluble finite groups are studied. Some earlier results on mutually permutable products of [Formula: see text]-supersoluble groups (SC-groups) appear as particular cases.

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.

scholarly journals On the structure of finite groups isospectral to finite simple groups

2015 ◽  
Vol 18 (5) ◽  
Author(s):  
Mariya A. Grechkoseeva ◽  
Andrey V. Vasil'ev
Keyword(s):  
Simple Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Element Orders ◽  
Nonabelian Simple Group

AbstractFinite groups are said to be isospectral if they have the same sets of element orders. A finite nonabelian simple group

scholarly journals Quasirecognition by Prime Graph of the Groups 2D2n(q) Where q < 105

2017 ◽  
Author(s):  
Hossein Moradi ◽  
Mohammad Reza Darafsheh ◽  
Ali Iranmanesh
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Prime Power ◽  
Prime Graph ◽  
Composition Factor ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Prime Divisors ◽  
A Finite Group

Let G be a finite group. The prime graph &Gamma;(G) of G is defined as follows: The set of vertices of&nbsp;&Gamma;(G) is the set of prime divisors of |G| and two distinct vertices p and p' are connected in &Gamma;(G), whenever G has an element of order pp'. A non-abelian simple group P is called recognizable by prime graph if for any finite group G with &Gamma;(G)=&Gamma;(P), G has a composition factor isomorphic to P. In&nbsp;[4] proved finite simple groups 2Dn(q), where&nbsp;n&nbsp;&ne; 4k are quasirecognizable by prime graph. Now in this paper we discuss the quasirecognizability by prime graph of the simple groups 2D2k(q), where&nbsp;k &ge; 9 and q is a prime power less than 105.

scholarly journals Steinberg-like characters for finite simple groups

2020 ◽  
Vol 23 (1) ◽  
pp. 25-78
Author(s):  
Gunter Malle ◽  
Alexandre Zalesski
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Simple Group ◽  
Regular Representation ◽  
Simple Groups ◽  
Closed Field ◽  
Finite Simple Groups ◽  
Algebraically Closed Field ◽  
Characteristic 2 ◽  
A Finite Group

AbstractLet G be a finite group and, for a prime p, let S be a Sylow p-subgroup of G. A character χ of G is called {\mathrm{Syl}_{p}}-regular if the restriction of χ to S is the character of the regular representation of S. If, in addition, χ vanishes at all elements of order divisible by p, χ is said to be Steinberg-like. For every finite simple group G, we determine all primes p for which G admits a Steinberg-like character, except for alternating groups in characteristic 2. Moreover, we determine all primes for which G has a projective FG-module of dimension {\lvert S\rvert}, where F is an algebraically closed field of characteristic p.

Constructing Representations of Finite Simple Groups and Covers

2006 ◽  
Vol 58 (1) ◽  
pp. 23-38 ◽  
Author(s):  
Vahid Dabbaghian-Abdoly
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Irreducible Character ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Simple Method ◽  
Covering Group ◽  
Representations Of Finite Groups ◽  
A Finite Group

AbstractLet G be a finite group and χ be an irreducible character of G. An efficient and simple method to construct representations of finite groups is applicable whenever G has a subgroup H such that χH has a linear constituent with multiplicity 1. In this paper we show (with a few exceptions) that if G is a simple group or a covering group of a simple group and χ is an irreducible character of G of degree less than 32, then there exists a subgroup H (often a Sylow subgroup) of G such that χH has a linear constituent with multiplicity 1.

scholarly journals ON HUPPERT’S CONJECTURE FOR THE CONWAY AND FISCHER FAMILIES OF SPORADIC SIMPLE GROUPS

2013 ◽  
Vol 94 (3) ◽  
pp. 289-303 ◽  
Author(s):  
S. H. ALAVI ◽  
A. DANESHKHAH ◽  
H. P. TONG-VIET ◽  
T. P. WAKEFIELD
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Simple Group ◽  
Irreducible Character ◽  
Character Degrees ◽  
Simple Groups ◽  
Nonabelian Simple Group ◽  
Sporadic Simple Groups ◽  
A Finite Group

AbstractLet $G$ denote a finite group and $\mathrm{cd} (G)$ the set of irreducible character degrees of $G$. Huppert conjectured that if $H$ is a finite nonabelian simple group such that $\mathrm{cd} (G)= \mathrm{cd} (H)$, then $G\cong H\times A$, where $A$ is an abelian group. He verified the conjecture for many of the sporadic simple groups and we complete its verification for the remainder.

scholarly journals Simple groups with the same prime graph as 2Dn(q)

2015 ◽  
Vol 98 (112) ◽  
pp. 251-263
Author(s):  
Behrooz Khosravi ◽  
A. Babai
Keyword(s):  
Positive Integer ◽  
Simple Group ◽  
Prime Graph ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Nonabelian Simple Group ◽  
Prime Graphs

In 2006, Vasil'ev posed the problem: Does there exist a positive integer k such that there are no k pairwise nonisomorphic nonabelian finite simple groups with the same graphs of primes? Conjecture: k = 5. In 2013, Zvezdina, confirmed the conjecture for the case when one of the groups is alternating. We continue this work and determine all nonabelian simple groups having the same prime graphs as the nonabelian simple group 2Dn(q).

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