On a group with three supersolvable subgroups of pairwise relatively prime indices

In this paper, we show that if G is a finite group with three supersolvable subgroups of pairwise relatively prime indices in G and G′ is nilpotent, then G is supersolvable. Let π(G) denote the set of prime divisors of |G| and max(π(G)) denote the largest prime divisor of |G|. We also establish that if G is a finite group such that G has three supersolvable subgroups H, K, and L whose indices in G are pairwise relatively prime, q \nmid p-1{q \nmid p-1} where p =  max(π(G)) and q = max(π(L)) with L a Hall p′-subgroup of G, then G is supersolvable.     

Primes dividing the degrees of the real characters

Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. The Ito–Michler Theorem asserts that if a prime p does not divide the degree of any χ Irr(G) then a Sylow p-subgroup P of G is normal in G. We prove a real-valued version of this theorem, where instead of Irr(G) we only consider the subset Irr_rv(G) consisting of all real-valued irreducible characters of G. We also prove that the character degree graph associated to Irr_rv(G) has at most 3 connected components. Similar results for the set of real conjugacy classes of G have also been obtained. Part of this paper was done while the second author visited the Mathematics Department of the Università di Firenze. He would like to thank the Department for its hospitality. The authors are also grateful to F. Lübeck for helping them with some computer calculations. The research of the first author was partially supported by MIUR research program “Teoria dei gruppi ed applicazioni”. This research of the second author was partially supported by the Spanish Ministerio de Educación y Ciencia proyecto MTM2004-06067-C02-01. The third author gratefully acknowledges the support of the NSA and the NSF.     

Pure quantitative characterization of finite simple groups

Let G be a finite group, and let π _e (G) be the set of all element orders of G. In this short paper we prove that π _e (B _n (q)) ≠ π _e (C _n (q)) for all odd q.      

On the relative character correspondences

Let π(G,A):Irr_A(G)→Irr(C_G(A)) be the Glauberman–Isaacs correspondence, where G and A are finite groups with coprime orders and A acts on G by automorphisms. Let B be a subgroup of A. In this setting, we give some new conditions for the fixed-point subgroups C_G(A) and C_G(B) such that χπ(G,A) is an irreducible constituent of the restriction of χπ(G,B) to C_G(A) for all χIrr_A(G).     

Verifying Huppert’s Conjecture for PSp<Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>) when <Emphasis Type="Italic">q</Emphasis> > 7

Let G denote a finite group and cd (G) the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd (G) = cd (H), then G ≅ H × A, where A is an abelian group. Huppert verified the conjecture for PSp₄(q) when q = 3, 4, 5, or 7. In this paper, we extend Huppert’s results and verify the conjecture for PSp₄(q) for all q. This demonstrates progress toward the goal of verifying the conjecture for all nonabelian simple groups of Lie type of rank two.     

Finite groups in whichp′-classes haveq′-length

IfG is a finite group in which every element ofp′-order centralizes aq-Sylow subgroup ofG, wherep andq are distinct primes, it is shown thatO ^q′ (G) is solvable,l _q (G)≤1 andl _p (O ^q′ (G)) ≤2. Further, the structure ofG is determined to some extent. Work partially supported by MURST research program “Teoria dei gruppi ed applicazioni”.     

Limq → ∞ \frac{c(G)}{r(G)} for Simple Groups

By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. For a given finite group G, let c(G) denote the minimal degree of a faithful representation of G by complex quasi-permutation matrices and let r(G) denote the minimal degree of a faithful rational valued character of G. Also let G denote one of the symbols A_l, B_l, C_l, D_l, E₆, E₇, E₈, G₂, F₄, ²B₂, ²E₄, ²G₂, and ³D₄. Let G(q) denote simple group of type G over GF(q). Let c(q) = c(G(q)) and r(q) = r(G(q)). Then we will show that lim Lim_q = 1.     

On r-recognition by prime graph of Bp(3) where p is an odd prime

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. We denote by k(Γ(G)) the number of isomorphism classes of finite groups H satisfying Γ(G) = Γ(H). Given a natural number r, a finite group G is called r-recognizable by prime graph if k(Γ(G)) =  r. In Shen et al. (Sib. Math. J. 51(2):244–254, 2010), it is proved that if p is an odd prime, then B _p (3) is recognizable by element orders. In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(B _p (3)), where p > 3 is an odd prime, then G @ B_p(3){G\cong B_p(3)} or C _p (3). Also if Γ(G) = Γ(B ₃(3)), then G @ B₃(3), C₃(3), D₄(3){G\cong B_3(3), C_3(3), D_4(3)}, or G/O₂(G) @ Aut(²B₂(8)){G/O_2(G)\cong {\rm Aut}(^2B_2(8))}. As a corollary, the main result of the above paper is obtained.     

On Recognizability of the Group E8(q) by the Set of Orders of Elements

We prove that if a finite group G has the same set of orders of elements as the group E ₈(q), then O ³(G/F(G)) is isomorphic to E ₈(q).     

On recognizability of some finite simple orthogonal groups by spectrum

It is proved that, if G is a finite group that has the same set of element orders as the simple group D _p (q), where p is prime, p ≥ 5 and q ∈ {2, 3, 5}, then the commutator group of G/F(G) is isomorphic to D _p (q), the subgroup F(G) is equal to 1 for q = 5 and to O _q (G) for q ∈ {2, 3}, F(G) ≤ G′, and |G/G′| ≤ 2.     

The phase transition in the cluster‐scaled model of a random graph

For 0 < p < 1 and q > 0 let G_q(n,p) denote the random graph with vertex set [n]={1,…,n} such that, for each graph G on [n] with e(G) edges and c(G) components, the probability that G_q(n,p)=G is proportional to . The first systematic study of G_q(n,p) was undertaken by 6 , who analyzed the phase transition phenomenon corresponding to the emergence of the giant component. In this paper we describe the structure of G_q(n,p) very close the critical threshold. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Sharp bounds for the number of 3‐independent partitions and the chromaticity of bipartite graphs

Given a graph G and an integer k ≥ 1, let α(G, k) denote the number of k‐independent partitions of G. Let ??^?s(p,q) (resp., ??₂^?s(p,q)) denote the family of connected (resp., 2‐connected) graphs which are obtained from the complete bipartite graph K_p,q by deleting a set of s edges, where p ≥ q ≥ 2. This paper first gives a sharp upper bound for α(G,3), where G ∈ ?? ^?s(p,q) and 0 ≤ s ≤ (p ? 1)(q ? 1) (resp., G ∈ ?? ₂^?s(p,q) and 0 ≤ s ≤ p + q ? 4). These bounds are then used to show that if G ∈ ?? ^?s(p,q) (resp., G ∈ ?? ₂^?s (p,q)), then the chromatic equivalence class of G is a subset of the union of the sets ??^?s_i(p+i,q?i) where max and s_i = s ? i(p?q+i) (resp., a subset of ??₂^?s(p,q), where either 0 ≤ s ≤ q ? 1, or s ≤ 2q ? 3 and p ≥ q + 4). By applying these results, we show finally that any 2‐connected graph obtained from K_p,q by deleting a set of edges that forms a matching of size at most q ? 1 or that induces a star is chromatically unique. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 48–77, 2001     

p-nilpotence of finite groups and minimal subgroups

In this paper, it is proved that a finite group G is p-nilpotent if every minimal subgroup of P∩O^p(G) is permutable in P and N_G(P) is p-nilpotent, and when p=2 either [Ω₂(P∩O^p(G)),P]Ω₁(P∩O^p(G)) or P is quaternion-free, where p is a prime dividing the order of G and P is a Sylow p-subgroup of G. By using this result, we may get a series of corollaries for p-nilpotence, which contain some known results. Some other applications of this result are also given.     

Special representations of the group SP(4,q) with minimal degrees

Let G be a finite group. Let p(G) denote the minimal degree of a faithful permutation representation ofG and let q(G) and c(G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices over the rational and the complex numbers, respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G. The purpose of this paper is to calculate p(G), q(G), c(G) and r(G) for the group SP(4,q). This revised version was published online in June 2006 with corrections to the Cover Date.     

The Asymptotic Distribution of Singular-Values with Applications to Canonical Correlations and Correspondence Analysis

Let X_n, n = 1, 2, ... be a sequence of p × q random matrices, p ≥ q. Assume that for a fixed p × q matrix B and a sequence of constants b_n → ∞, the random matrix b_n(X_n − B) converges in distribution to Z. Let ψ(X_n) denote the q-vector of singular values of X_n. Under these assumptions, the limiting distribution of b_n (ψ(X_n) − ψ(B)) is characterized as a function of B and of the limit matrix Z. Applications to canonical correlations and to correspondence analysis are given.     

Above the Glauberman correspondence

Let S be a finite solvable group, and suppose S acts on the finite group N, and they have coprime orders. Then, the celebrated Glauberman correspondence provides a natural bijection from the set IrrS(N) of irreducible characters of N which are invariant under the action of S to the set Irr(CN(S)) of all irreducible characters of the centralizer of S in N. Suppose, further, that the semidirect product SN is a normal subgroup of a finite group G. Let θ∈IrrS(N), and let ψ∈Irr(CN(S)) be its Glauberman correspondent. We prove that there is a bijection with good compatibility properties from the set Irr(G,θ) of the irreducible characters of G above θ to Irr(NG(S),ψ) such that, in the case when S is a p-group for some prime p, it preserves fields of values and Schur indices over Qp, the field of p-adic numbers. Using this result, we also prove a strengthening of the McKay Conjecture for all p-solvable groups.     

Simple exceptional groups of Lie type are determined by their character degrees

Let G be a finite group. Denote by Irr(G) the set of all irreducible complex characters of G. Let cd(G)={c(1)  |  c ? Irr(G)}{{\rm cd}(G)=\{\chi(1)\;|\;\chi\in {\rm Irr}(G)\}} be the set of all irreducible complex character degrees of G forgetting multiplicities, and let X₁(G) be the set of all irreducible complex character degrees of G counting multiplicities. Let H be any non-abelian simple exceptional group of Lie type. In this paper, we will show that if S is a non-abelian simple group and cd(S) í cd(H){{\rm cd}(S)\subseteq {\rm cd}(H)} then S must be isomorphic to H. As a consequence, we show that if G is a finite group with X₁(G) í X₁(H){{\rm X}_1(G)\subseteq {\rm X}_1(H)} then G is isomorphic to H. In particular, this implies that the simple exceptional groups of Lie type are uniquely determined by the structure of their complex group algebras.     

Normal <Emphasis Type="Italic">π</Emphasis>-complements for finite groups

Let G be a finite group. For a finite p-group P the subgroup generated by all elements of order p is denoted by Ω₁(p). Zhang [5] proved that if P is a Sylow p-subgroup of G, Ω₁(P) ≦ Z(P) and N _G (Z(P)) has a normal p-complement, then G has a normal p-complement. The object of this paper is to generalize this result. This paper was partly supported by Hungarian National Foundation for Scientific Research Grant # T049841 and T038059.     

On Thompson's Theorem

Let p be a prime divisor of the order of a finite group G. Thompson (1970, J. Algebra14, 129–134) has proved the following remarkable result: a finite group G is p-nilpotent if the degrees of all its nonlinear irreducible characters are divisible by p (in fact, in that case G is solvable). In this note, we prove that a group G, having only one nonlinear irreducible character of p′-degree is a cyclic extension of Thompson's group. This result is a consequence of the following theorem: A nonabelian simple group possesses two nonlinear irreducible characters χ₁ and χ₂ of distinct degrees such that p does not divide χ₁(1)χ₂(1) (here p is arbitrary but fixed). Our proof depends on the classification of finite simple groups. Some properties of solvable groups possessing exactly two nonlinear irreducible characters of p′-degree are proved. Some open questions are posed.     

A Family of Finite Simple Groups Which Are 2-Recognizable by Their Elements Order

Abstract

Let G be a finite group and ω(G) the set of all orders of elements in G. Denote by h(ω(G)) the number of isomorphism classes of finite groups H satisfying ω(H) = ω(G), and put h(G) = h(ω(G)). A group G is called k-recognizable if h(G) = k < ∞ , otherwise G is called non-recognizable. In the present article we will show that the simple groups PSL(3, q), where q ≡ ±2(mod 5) and (6, (q ? 1)/2) = 2, are 2-recognizable. Therefore if q is a prime power and q ≡ 17, 33, 53 or 57 (mod 60), then the groups PSL(3, q) are 2-recognizable. Hence proving the existing of an infinite families of 2-recognizable simple groups.