Thompson's Conjecture Research Articles

Corrigendum and addendum to “Centralizers of finite subgroups in Hall’s universal group”

In Hall's universal group every non-trivial conjugacy class satisfies CC=U . Hence generalized version of J. G. Thompson's conjecture is true for every non-trivial conjugacy class C in U . Moreover Ore's conjecture (every element is a commutator) is true for U is added to [4]. In [4, Theorem 2.4] C_U(F)/Z(F)\cong U is true if Z(F)=1 .

The influence of order and conjugacy class length on the structure of finite groups

Let $2^{n}+1 \gt 5$ be a prime number. In this article, we will show $G\cong C_{n}(2)$ if and only if $|G|=|C_{n}(2)|$ and $G$ has a conjugacy class length ${|C_{n}(2)|}/({2^{n}+1})$. Furthermore, we will show Thompson's conjecture is valid under a weak condition for the symplectic groups $C_{n}(2)$.

Improving Thompson's Conjecture for Suzuki Groups

Let G be a finite group and cs(G) be the set of conjugacy class sizes of G. In 1987, J. G. Thompson conjectured that, if G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying that cs(G) = cs(M), then G ≅ M. This conjecture has been proved for Suzuki groups in [5]. In this article, we improve this result by proving that, if G is a finite group such that cs(G) = cs(Sz(q)), for q = 22m+1, then G ≅ Sz(q) × A, where A is abelian. We avoid using classification of finite simple groups in our proofs.

Towards Thompson's conjecture for alternating and symmetric groups

Abstract For a finite group G, let N(G) denote the set of conjugacy class sizes of G. We show that if G is a finite group such that N(G) = N(Alt n ) for n > 4, or N(G) = N(Sym n ) for n > 22, then G is non-solvable.

Recognizing alternating groups by their order and one conjugacy class length

In this paper, the alternating groups of degree p are characterized by their order and one conjugacy class length, where p is a prime number. This work implies that Thompson's conjecture holds for the alternating groups of degree p.

On the Validity of Thompson's Conjecture for Finite Simple Groups

In this article, we prove a conjecture of J. G. Thompson for the finite simple group 2 D n (q). More precisely, we show that every finite group G with the property Z(G) = 1 and N(G) = N(2 D n (q)) is necessarily isomorphic to 2 D n (q). Note that N(G) is the set of lengths of conjugacy classes of G.

ON THOMPSON'S CONJECTURE FOR ALMOST SPORADIC SIMPLE GROUPS

Let G be a non-abelian group and N(G) be the set of conjugacy class sizes of G. In 1980s, Thompson posed the following conjecture: if M is a finite non-abelian simple group, G is a finite group with Z(G) = 1 and N(G) = N(M), then M ≅ G. Here, we prove Thompson's conjecture holds for all almost sporadic simple groups.

CONJUGATE GRAPHS OF FINITE GROUPS

In this paper we introduce the conjugate graph [Formula: see text] associated to a nonabelian group G with vertex set G\Z(G) such that two distinct vertices join by an edge if they are conjugate. We show if [Formula: see text], where S is a finite nonabelian simple group which satisfy Thompson's conjecture, then G ≅ S. Further, if central factors of two nonabelian groups H and G are isomorphic and |Z(G)| = |Z(H)|, then H and G have isomorphic conjugate graphs.

On Thompson's Conjecture of A 10

In this short article, we prove the Thompson's Conjecture is true for the alternating group A 10, where A 10 has exactly one connected prime graph component.

CHARACTERIZATION OF THE GROUPS Dp+1(2) AND Dp+1(3) USING ORDER COMPONENTS

In this paper we will prove that the groups <TEX>$D_{p+1}$</TEX>(2) and <TEX>$D_{p+1}$</TEX>(3), where p is an odd prime number, are uniquely determined by their sets of order components. A main consequence of our result is the validity of Thompson's conjecture for the groups <TEX>$D_{p+1}$</TEX>(2) and <TEX>$D_{p+1}$</TEX>(3).

A New Characterization of A 10 by its Noncommuting Graph

Let G be a nonabelian group and associate a noncommuting graph ∇(G) with G as follows: The vertex set of ∇(G) is G\\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, Professor J. G. Thompson gave the following conjecture. Thompson's Conjecture. If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ≅ M, where N(G):={n ∈ ℕ | G has a conjugacy class of size n}. In 2006, A. Abdollahi, S. Akbari, and H. R. Maimani put forward a conjecture (AAM's conjecture) in Abdollahi et al. (2006) as follows. AAM's Conjecture. Let M be a finite nonabelian simple group and G a group such that ∇(G) ≅ ∇ (M). Then G ≅ M. In this short article we prove that if G is a finite group with ∇(G) ≅ ∇ (A 10), then G ≅ A 10, where A 10 is the alternating group of degree 10.

A new characterization of alternating and symmetric groups

In this paper we prove that the alternating groupsAn, forn=p, p+1, p+2 and symmetric groupsSn, forn=p, p+1, wherep>=3 is a prime number, can be uniquely determined by their order components. As one of the important consequence of this characterization we show that the simple groupsAn, wheren=p, p+1, p+2 andp>=3 is prime, satisfy in Thompson's conjecture and Shi's conjecture.

A New Characterization of PSL(p, q)#

Order components of a finite group are introduced in Chen [Chen, G. Y. (1996c) On Thompson's conjecture. J. Algebra 185:184–193]. It was proved that PSL(3, q), where q is an odd prime power, is uniquely determined by its order components [Iranmanesh, A., Alavi, S. H., Khosravi, B. (2002a). A characterization of PSL(3, q) where q is an odd prime power. J. Pure Appl. Algebra 170(2–3): 243–254]. Also in Iranmanesh et al. [Iranmanesh, A., Alavi, S. H., Khosravi, B. (2002b). A characterization of PSL(3, q) where q = 2 n . Acta Math. Sinica, English Ser. 18(3):463–472] and [Iranmanesh, A., Alavi, S. H. (2002). A characterization of simple groups PSL(5, q). Bull. Austral. Math. Soc. 65:211–222] it was proved that PSL(3, q) for q = 2 n and PSL(5, q) are uniquely determined by their order components. In this paper we prove that PSL(p, q) can be uniquely determined by its order components, where p is an odd prime number. A main consequence of our results is the validity of Thompson's conjecture for the groups under consideration.

A characterization of PSL(3, q) where q is an odd prime power

Order components of a finite group are introduced in Chen (J. Algebra 15 (1996) 184). We prove that PSL(3, q) can be uniquely determined by its order components where q is an odd prime power. A main consequence of our result is the validity of Thompson's conjecture for the groups under consideration.

A characterisation of simple groupsPSL(5,q)

Order components of a finite group are introduced in [4]. We prove that, for everyq,PSL(5,q) can be uniquely determined by its order componets. A main consequence of our result is the validity of Thompson's conjecture for the groups under consideration.

Random walks on symmetric spaces and inequalities for matrix spectra

Using harmonic analysis on symmetric spaces we reduce the singular spectral problem for products of matrices to the recently solved spectral problem for sums of Hermitian matrices. This proves R.C. Thompson's conjecture [Matrix Spectral Inequalities, Johns Hopkins University Press, Baltimore, MD, 1988].

A matrix-decomposition theorem for GL n( K)

Given an arbitrary commutative field K, n∈ N ⩾3 and two monic polynomials q and r over K of degree n−1 and n such that q(0)≠0≠ r(0). We prove that any non-scalar invertible n×n matrix M can be written as a product of two matrices A and B, where the minimum polynomial of A is divisible by q and B is cyclic with minimum polynomial r. This result yields that the Thompson conjecture is true for PSL n( F 3) , n∈ N ⩾3 , and PSL 2n+1( F 2) , n∈ N . If G is such a group, then G has a conjugacy class Ω such that G=Ω 2 . In particular each element of G is a commutator.

Further Reflections on Thompson's Conjecture

Let G be a finite group and let N(G)={n∈N|G has a conjugacy class C, such that |C|=n}. Professor J. G. Thompson has conjectured that: If G is a finite group with Z(G)=1 and M a non-abelian simple group satisfying N(G)=N(M), then G≅M.We have proved previously that: If M is a sporadic simple group or a simple group having its prime graph with at least three prime graph components, then Thompson's conjecture is correct. In this paper, we shall prove:Main Theorem. Let G be a finite group with Z(G)=1 and M=G2(q) or G2(2)′, where q≥2, such that N(G)=N(M). Then G≅M.

On Thompson's Conjecture

LetGbe a finite group andN(G)={n∈N|Ghas a conjugacy classC, such that |C|=n}. Professor J. G. Thompson has conjectured that “IfGbe a finite group withZ(G)=1 andMa nonabelian simple group satisfying thatN(G)=N(M), thenG≅M.” We have proved that ifMis a sporadic simple group, then Thompson's conjecture is correct. In this paper, we shall further prove that ifMis a finite simple group having at least three prime graph components, then the conjecture is also correct.