A new characterization for some linear groups

Let G be a group and π _e (G) be the set of element orders of G. Let k ? p_e(G){k\in\pi_e(G)} and m _k be the number of elements of order k in G. Let nse(G) = {m_k|k ? p_e(G)}{{\rm nse}(G) = \{m_k|k\in\pi_e(G)\}} . In Shen et al. (Monatsh Math, 2009), the authors proved that A₄ @ PSL(2, 3), A₅ @ PSL(2, 4) @ PSL(2,5){A_4\cong {\rm PSL}(2, 3), A_5\cong \rm{PSL}(2, 4)\cong \rm{PSL}(2,5)} and A₆ @ PSL(2,9){A_6\cong \rm{PSL}(2,9)} are uniquely determined by nse(G). In this paper, we prove that if G is a group such that nse(G) = nse(PSL(2, q)), where q ? {7,8,11,13}{q\in\{7,8,11,13\}} , then G @ PSL(2,q){G\cong {PSL}(2,q)} .     

Characterization of $ {\mathbb{A}_{16}} $ by a noncommuting graph

Let G be a finite non-Abelian group. We define a graph Γ _G ; called the noncommuting graph of G; with a vertex set G − Z(G) such that two vertices x and y are adjacent if and only if xy ≠ yx: Abdollahi, Akbari, and Maimani put forward the following conjecture (the AAM conjecture): If S is a finite non-Abelian simple group and G is a group such that Γ _S ≅ Γ _G ; then S ≅ G: It is still unknown if this conjecture holds for all simple finite groups with connected prime graph except \mathbbA₁₀ {\mathbb{A}_{10}} , L ₄(8), L ₄(4), and U ₄(4). In this paper, we prove that if \mathbbA₁₆ {\mathbb{A}_{16}} denotes the alternating group of degree 16; then, for any finite group G; the graph isomorphism G_\mathbbA16 @ G_G {\Gamma_{{\mathbb{A}_{16}}}} \cong {\Gamma_G} implies that \mathbbA₁₆ @ G {\mathbb{A}_{16}} \cong G .     

Quasirecognition by prime graph of F4(q) where q?=?2n?>?2

The prime graph of a finite group G is denoted by Γ(G). In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(F ₄(q)), where q = 2 ⁿ  > 2, then G has a unique nonabelian composition factor isomorphic to F ₄(q). We also show that if G is a finite group satisfying |G| = |F ₄(q)| and Γ(G) = Γ(F ₄(q)), where q = 2 ⁿ  > 2, then G @ F₄(q){G \cong F_4(q)}. As a consequence of our result we give a new proof for a conjecture of Shi and Bi for F ₄(q) where q = 2 ⁿ  > 2.     

The intersection of subgroups of finite <Emphasis Type="Italic">p</Emphasis>-groups

For a finite p-group G and a positive integer k let I _k (G) denote the intersection of all subgroups of G of order p ^k . This paper classifies the finite p-groups G with I_k(G) @ C_p^k-1{{I}_k(G)\cong C_{p^{k-1}}} for primes p > 2. We also show that for any k, α ≥ 0 with 2(α + 1) ≤ k ≤ n−α the groups G of order p ⁿ with I_k(G) @ C_p^k-a{{I}_k(G)\cong C_{p^{k-\alpha}}} are exactly the groups of exponent p ^n-α .     

On the prime graph of <Emphasis Type="Italic">PSL</Emphasis>(2, <Emphasis Type="Italic">p</Emphasis>) where <Emphasis Type="Italic">p</Emphasis> > 3 is a prime number

Let G be a finite group. We define the prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. Recently M. Hagie [5] determined finite groups G satisfying Γ(G) = Γ(S), where S is a sporadic simple group. Let p > 3 be a prime number. In this paper we determine finite groups G such that Γ(G) = Γ(PSL(2, p)). As a consequence of our results we prove that if p > 11 is a prime number and p ≢ 1 (mod 12), then PSL(2, p) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph. The third author was supported in part by a grant from IPM (No. 84200024).     

Recognition by prime graph of $^2 D_{2^m + 1} (3)$

As shown in [1] the simple group ² D_{2^m + 1} (3)^2 D_{2^m + 1} (3) is recognizable by spectrum. The main result of this paper generalizes the above, stating that ² D_{2^m + 1} (3)^2 D_{2^m + 1} (3) is recognizable by prime graph. In other words, we show that if G is a finite group satisfying G(G) = G(² D_{2^m + 1} (3))\Gamma (G) = \Gamma (^2 D_{2^m + 1} (3)) then G @ ² D_{2^m + 1} (3)G \cong ^2 D_{2^m + 1} (3).     

A Family of Nonnormal Cayley Digraphs

We call a Cayley digraph Γ = Cay(G, S) normal for G if G _R , the right regular representation of G, is a normal subgroup of the full automorphism group Aut(Γ) of Γ. In this paper we determine the normality of Cayley digraphs of valency 2 on nonabelian groups of order 2p ² (p odd prime). As a result, a family of nonnormal Cayley digraphs is found. Received February 23, 1998, Revised September 25, 1998, Accepted October 27, 1998     

On <Emphasis Type="Italic">r</Emphasis>-recognition by prime graph of <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(3) where <Emphasis Type="Italic">p</Emphasis> 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 \({G\cong B_p(3)}\) or C _p (3). Also if Γ(G) = Γ(B ₃(3)), then \({G\cong B_3(3), C_3(3), D_4(3)}\), or \({G/O_2(G)\cong {\rm Aut}(^2B_2(8))}\). As a corollary, the main result of the above paper is obtained.     

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.     

Groups whose vanishing class sizes are not divisible by a given prime

Let G be a finite group. An element ${g\in G}Let G be a finite group. An element g ? G{g\in G} is a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g) = 0: if this is the case, we say that the conjugacy class of g in G is a vanishing conjugacy class of G. In this paper we show that, if the size of every vanishing conjugacy class of G is not divisible by a given prime number p, then G has a normal p-complement and abelian Sylow p-subgroups.     

On a graph related to permutability in finite groups

For a finite group G we define the graph Γ(G) to be the graph whose vertices are the conjugacy classes of cyclic subgroups of G and two conjugacy classes ${\mathcal {A}, \mathcal {B}}For a finite group G we define the graph Γ(G) to be the graph whose vertices are the conjugacy classes of cyclic subgroups of G and two conjugacy classes A, B{\mathcal {A}, \mathcal {B}} are joined by an edge if for some A ? A, B ? B A{A \in \mathcal {A},\, B \in \mathcal {B}\, A} and B permute. We characterise those groups G for which Γ(G) is complete.     

Metric properties of the tropical Abel–Jacobi map

Let Γ be a tropical curve (or metric graph), and fix a base point p∈Γ. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(Γ) is canonically isomorphic to the direct limit of J(G) over all weighted graph models G for Γ. This result is useful for reducing certain questions about the Abel–Jacobi map Φ _p :Γ→J(Γ), defined by Mikhalkin and Zharkov, to purely combinatorial questions about weighted graphs. We prove that J(G) is finite if and only if the edges in each 2-connected component of G are commensurable over ℚ. As an application of our direct limit theorem, we derive some local comparison formulas between ρ and \varPhi_p^*(r){\varPhi}_{p}^{*}(\rho) for three different natural “metrics” ρ on J(Γ). One of these formulas implies that Φ _p is a tropical isometry when Γ is 2-edge-connected. Another shows that the canonical measure μ _Zh on a metric graph Γ, defined by S. Zhang, measures lengths on Φ _p (Γ) with respect to the “sup-norm” on J(Γ).     

Vertex-Distinguishing Edge Colorings of Graphs with Degree Sum Conditions

An edge coloring is called vertex-distinguishing if every two distinct vertices are incident to different sets of colored edges. The minimum number of colors required for a vertex-distinguishing proper edge coloring of a simple graph G is denoted by c￠_vd(G){\chi'_{vd}(G)}. It is proved that c￠_vd(G) ￡ D(G)+5{\chi'_{vd}(G)\leq\Delta(G)+5} if G is a connected graph of order n ≥ 3 and s₂(G) 3 \frac2n3{\sigma_{2}(G)\geq\frac{2n}{3}}, where σ ₂(G) denotes the minimum degree sum of two nonadjacent vertices in G.     

Groups with the same prime graph as the orthogonal group B n (3)

Let G be a finite group. The prime graph of G is denoted by Γ(G). It is proved in [1] 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) or C _p (3). In this paper we prove the main result that if G is a finite group such that Γ(G) = Γ(B _n (3)), where n ≥ 6, then G has a unique nonabelian composition factor isomorphic to B _n (3) or C _n (3). Also if Γ(G) = Γ(B ₄(3)), then G has a unique nonabelian composition factor isomorphic to B ₄(3), C ₄(3), or ² D ₄(3). It is proved in [2] that if p is an odd prime, then B _p (3) is recognizable by element orders. We give a corollary of our result, generalize the result of [2], and prove that B _2k+1(3) is recognizable by the set of element orders. Also the quasirecognition of B _2k (3) by the set of element orders is obtained.     

Connected Domination Number of a Graph and its Complement

A set S of vertices in a graph G is a connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraph induced by S is connected. The connected domination number γ _c (G) is the minimum size of such a set. Let d^*(G)=min{d(G),d([`(G)])}{\delta^*(G)={\rm min}\{\delta(G),\delta({\overline{G}})\}} , where [`(G)]{{\overline{G}}} is the complement of G and δ(G) is the minimum vertex degree. We prove that when G and [`(G)]{{\overline{G}}} are both connected, g<sub>c</sub>(G)+g<sub>c</sub>([`(G)]) ￡ d^*(G)+4-(g_c(G)-3)(g_c([`(G)])-3){{\gamma_c}(G)+{\gamma_c}({\overline{G}})\le \delta^*(G)+4-({\gamma_c}(G)-3)({\gamma_c}({\overline{G}})-3)} . As a corollary, g<sub>c</sub>(G)+g<sub>c</sub>([`(G)]) ￡ \frac3n4{{\gamma_c}(G)+{\gamma_c}({\overline{G}})\le \frac{3n}{4}} when δ*(G) ≥ 3 and n ≥ 14, where G has n vertices. We also prove that g_c(G)+g_c([`(G)]) ￡ d<sup>*</sup>(G)+2{{\gamma_c}(G)+{\gamma_c}({\overline{G}})\le \delta^*(G)+2} when g<sub>c</sub>(G),g<sub>c</sub>([`(G)]) 3 4{{\gamma_c}(G),{\gamma_c}({\overline{G}})\ge 4} . This bound is sharp when δ*(G) = 6, and equality can only hold when δ*(G) = 6. Finally, we prove that g_c(G)g_c([`(G)]) ￡ 2n-4{{\gamma_c}(G){\gamma_c}({\overline{G}})\le 2n-4} when n ≥ 7, with equality only for paths and cycles.     

On minimal p-degrees in 2-transitive permutation groups

Suppose G is a transitive permutation group on a finite set W\mit\Omega of n points and let p be a prime divisor of |G||G|. The smallest number of points moved by a non-identity p-element is called the minimal p-degree of G and is denoted m_p (G). ¶ In the article the minimal p-degrees of various 2-transitive permutation groups are calculated. Using the classification of finite 2-transitive permutation groups these results yield the main theorem, that m_p(G) 3 [(p-1)/(p+1)] ·|W|m_{p}(G) \geq {{p-1} \over {p+1}} \cdot |\mit\Omega | holds, if Alt(W) \nleqq G {\rm Alt}(\mit\Omega ) \nleqq G .¶Also all groups G (and prime divisors p of |G||G|) for which m_p(G) ￡ [(p-1)/(p)] ·|W|m_{p}(G)\le {{p-1}\over{p}} \cdot |\mit\Omega | are identified.     

A Property of Isometric Mappings Between Dual Polar Spaces of Type {DQ (2n, {\mathbb K})}

Let f be an isometric embedding of the dual polar space ${\Delta = DQ(2n, {\mathbb K})}Let f be an isometric embedding of the dual polar space D = DQ(2n, \mathbb K){\Delta = DQ(2n, {\mathbb K})} into D￠ = DQ(2n, \mathbb K￠){\Delta^\prime = DQ(2n, {\mathbb K}^\prime)}. Let P denote the point-set of Δ and let e￠: D￠? S￠ @ PG(2ⁿ - 1, \mathbb K￠){e^\prime : \Delta^\prime \rightarrow {\Sigma^\prime} \cong {\rm PG}(2^n - 1, {{\mathbb K}^\prime})} denote the spin-embedding of Δ′. We show that for every locally singular hyperplane H of Δ, there exists a unique locally singular hyperplane H′ of Δ′ such that f(H) = f(P) ?H￠{f(H) = f(P) \cap H^\prime}. We use this to show that there exists a subgeometry S @ PG(2ⁿ - 1, \mathbb K){\Sigma \cong {\rm PG}(2^n - 1, {\mathbb K})} of Σ′ such that: (i) e￠°f (x) ? S{e^\prime \circ f (x) \in \Sigma} for every point x of D; (ii) e : = e￠°f{\Delta; ({\rm ii})\,e := e^\prime \circ f} defines a full embedding of Δ into Σ, which is isomorphic to the spin-embedding of Δ.     

A Generalization of the prime geodesic theorem to counting conjugacy classes of free subgroups

The classical prime geodesic theorem (PGT) gives an asymptotic formula (as x tends to infinity) for the number of closed geodesics with length at most x on a hyperbolic manifold M. Closed geodesics correspond to conjugacy classes of π₁(M) = Γ where Γ is a lattice in G = SO(n,1). The theorem can be rephrased in the following format. Let be the space of representations of into Γ modulo conjugation by Γ. is defined similarly. Let be the projection map. The PGT provides a volume form vol on such that for sequences of subsets {B _t }, satisfying certain explicit hypotheses, |π⁻¹(B _t )| is asymptotic to vol(B _t ). We prove a statement having a similar format in which is replaced by a free group of finite rank under the additional hypothesis that n = 2 or 3.      

A Spectral Gap Property for Random Walks Under Unitary Representations

Let G be a locally compact group and μ a probability measure on G, which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation $\pi ,\mathcal{H}Let G be a locally compact group and μ a probability measure on G, which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation p,H\pi ,\mathcal{H} of G, we study spectral properties of the operator π(μ) acting on H\mathcal{H} Assume that μ is adapted and that the trivial representation 1 _G is not weakly contained in the tensor product p?[`(p)]\pi\otimes \overline\pi We show that π(μ) has a spectral gap, that is, for the spectral radius r_spec(p(m))r_{\rm spec}(\pi(\mu)) of π(μ), we have r_spec(p(m)) < 1.r_{\rm spec}(\pi(\mu))< 1. This provides a common generalization of several previously known results. Another consequence is that, if G has Kazhdan’s Property (T), then r_spec(p(m)) < 1r_{\rm spec}(\pi(\mu))< 1 for every unitary representation π of G without finite dimensional subrepresentations. Moreover, we give new examples of so-called identity excluding groups.     

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.