A class of Hamiltonian and edge symmetric Cayley graphs on symmetric groups

Abstract. Let Sn be the symmetric group     

Finite groups with 24 elements of maximal order

It is an interesting topic to determine the structure of a finite group with a given number of elements of maximal order. In this article, we classify finite groups with 24 elements of maximal order.     

Recognition of finite groups by the prime graph

We obtain the first example of an infinite series of finite simple groups that are uniquely determined by their prime graph in the class of all finite groups. We also show that there exist almost simple groups for which the number of finite groups with the same prime graph is equal to 2. Supported by RFBR grant No. 05-01-00797, and by SB RAS Young Researchers Support grant No. 29 and Integration project No. 2006.1.2. __________ Translated from Algebra i Logika, Vol. 45, No. 4, pp. 390–408, July–August, 2006.     

A characterization of alternating groups by the set of orders of maximal abelian subgroups

We prove that alternating groups with three prime graph components are uniquely determined by the set of orders of maximal abelian subgroups.     

The order of elements in Sylow p-subgroups of the symmetric group

Define a random variable ξ_n by choosing a conjugacy class C of the Sylow p-subgroup of S_pn by random, and let ξ_n be the logarithm of the order of an element in C. We show that ξ_n has bounded variance and mean order log n /log p +O(1), which differs greatly from the average order of elements chosen with equal probability. This revised version was published online in June 2006 with corrections to the Cover Date.     

Generalized symmetric elements generated by a prime sequence

Let be a prime sequence in a local Noether lattice L. For denotes the set of finite joins in L of power products of the generalized symmetric elements of order k (majorization elements) in together with 0 and I. We have previously showed that for is a Noetherian distributive -domain. For and for any is again such a sub--domain of . For and is not closed under the meet of . However with its induced meet is again a Noetherian distributive -domain. Each finite set of majorization elements asymptotically forms a distributive sublattice of for k sufficiently large. Received March 2, 1998; accepted in final form June 11, 1998.     

素数立方阶群局部传递的图

目的是研究局部传递图的性质和分类.运用置换群和陪集图的理论,获得了关于素数立方阶群局部传递图的完全分类,证明了这些图是一些互不相交的关于素数立方阶群边传递图的并.     

OD-Characterization of alternating and symmetric groups of degrees 16 and 22

Let G be a finite group and π(G) be the set of all prime divisors of its order. The prime graph GK(G) of G is a simple graph with vertex set π(G), and two distinct primes p, q ∈ π(G) are adjacent by an edge if and only if G has an element of order pq. For a vertex p ∈ π(G), the degree of p is denoted by deg(p) and as usual is the number of distinct vertices joined to p. If π(G) = {p ₁, p ₂,...,p _k }, where p ₁ < p ₂ < ... < p _k , then the degree pattern of G is defined by D(G) = (deg(p ₁), deg(p ₂),...,deg(p _k )). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions |H| = |G| and D(H) = D(G). In addition, a 1-fold OD-characterizable group is simply called OD-characterizable. In the present article, we show that the alternating group A ₂₂ is OD-characterizable. We also show that the automorphism groups of the alternating groups A ₁₆ and A ₂₂, i.e., the symmetric groups S ₁₆ and S ₂₂ are 3-fold OD-characterizable. It is worth mentioning that the prime graph associated to all these groups are connected.     

Recognition by spectrum for finite simple groups of Lie type

The goal of this article is to survey new results on the recognition problem. We focus our attention on the methods recently developed in this area. In each section, we formulate related open problems. In the last two sections, we review arithmetical characterization of spectra of finite simple groups and conclude with a list of groups for which the recognition problem was solved within the last three years.      

Action of generalized symmetric groups on symmetric and exterior powers of their natural representations

We establish upper and lower bounds on the dimension of the space spanned by the symmetric powers of the natural character of generalized symmetric groups. We adapt the methods of Savitt and Stanley from [4 Savitt, D., Stanley, R. P. (2000). A note on the symmetric powers of the standard representation of S_n. Electron. J. Combin. 7:R6. [Google Scholar]] to obtain bounds both over the complex numbers and in prime characteristic.     

Recognition of some families of finite simple groups by order and set of orders of vanishing elements

Let G be a finite group. An element g ∈ G is called a vanishing element if there exists an irreducible complex character χ of G such that χ(g)= 0. Denote by Vo(G) the set of orders of vanishing elements of G. Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let G be a finite group and M a finite nonabelian simple group such that Vo(G) = Vo(M) and |G| = |M|. Then G ≌ M. We answer in affirmative this conjecture for M = Sz(q), where q = 2²ⁿ⁺¹ and either q ? 1, \(q - \sqrt {2q} + 1\) or q + \(\sqrt {2q} + 1\) is a prime number, and M = F⁴(q), where q = 2 ⁿ and either q⁴ + 1 or q⁴ ? q² + 1 is a prime number.     

Groups with the same order and degree pattern

The degree pattern of a finite group has been introduced in [18].A group M is called k-fold OD- characterizable if there exist exactly k non-isomorphic finite groups having the same order and degree pattern as M .In particular,a 1-fold OD-characterizable group is simply called OD-characterizable.It is shown that the alternating groups A m and A m+1 ,for m = 27,35,51,57,65,77,87,93 and 95,are OD-characterizable,while their automorphism groups are 3-fold OD-characterizable.It is also shown that the symmetric groups S m+2 ,for m = 7,13,19,23,31,37,43,47,53,61,67,73,79,83,89 and 97,are 3-fold OD-characterizable.From this,the following theorem is derived.Let m be a natural number such that m 100.Then one of the following holds: (a) if m = 10,then the alternating groups A m are OD-characterizable,while the symmetric groups S m are OD- characterizable or 3-fold OD-characterizable;(b) the alternating group A 10 is 2-fold OD-characterizable;(c) the symmetric group S 10 is 8-fold OD-characterizable.This theorem completes the study of OD-characterizability of the alternating and symmetric groups A m and S m of degree m 100.     

Characterizability of the Group ^2Dp（3） by its Order Components,Where p ≥ 5 is a Prime Number Not of the Form 2^m ＋ 1

The author will prove that the group ^2Dp（3） can be uniquely determined by its order components, where p ≠ 2^m ＋ 1 is a prime number, p ≥ 5. More precisely, if OC（G） denotes the set of order components of G, we will prove OC（G） = OC（^2Dp（3）） if and only if G is isomorphic to ^2Dp（3）. A main consequence of our result is the validity of Thompson＇s conjecture for the groups under consideration.     

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 spectrum of the groups 2~D_(2m+1)(3)

It is proved that a finite group with the same set of element orders as the simple group 2~D_(2m+1)(3) is isomorphic to 2~D_(2m+1)(3).     

Recognition of finite groups by a set of orders of their elements

For G a finite group, ω(G) denotes the set of orders of elements in G. If ω is a subset of the set of natural numbers, h(ω) stands for the number of nonisomorphic groups G such that ω(G)=ω. We say that G is recognizable (by ω(G)) if h(ω(G))=1. G is almost recognizable (resp., nonrecognizable) if h(ω(G)) is finite (resp., infinite). It is shown that almost simple groups PGL_n(q) are nonrecognizable for infinitely many pairs (n, q). It is also proved that a simple group S₄(7) is recognizable, whereas A₁₀, U₃(5), U₃(7), U₄(2), and U₅(2) are not. From this, the following theorem is derived. Let G be a finite simple group such that every prime divisor of its order is at most 11. Then one of the following holds: (i) G is isomorphic to A₅, A₇, A₈, A₉, A₁₁, A₁₂, L₂(q), q=7, 8, 11, 49, L₃(4), S₄(7), U₄(3), U₆(2), M₁₁, M₁₂, M₂₂, HS, or M^cL, and G is recognizable by the set ω(G); (ii) G is isomorphic to A₆, A₁₀, U₃(3), U₄(2), U₅(2), U₃(5), or J₂, and G is nonrecognizable; (iii) G is isomorphic to S₆(2) or O ₈ ⁺ (2), and h(ω(G))=2. Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 651–666, November–December, 1998.     

The local finiteness of some groups generated by a conjugacy class of order 3 elements

We prove local finiteness for the groups generated by a conjugacy class of order 3 elements whose every pair generates a subgroup that is isomorphic to Z ₃, A ₄, A ₅, SL ₂(3), or SL ₂(5).     

Recognition by spectrum of the groups $$ ^2 D_{2^m + 1} (3) $$

Abstract It is proved that a finite group with the same set of element orders as the simple group is isomorphic to . This work was supported by Russian Foundation for Basic Research (Grant No. 07-01-00148), RFBR-BRFBR (Grant No. 08-01-90006) and RFBR-GFEN (Grant No. 08-01-92200)     

A classification of all symmetric block designs of order nine with an automorphism of order six

We complete the classification of all symmetric designs of order nine admitting an automorphism of order six. As a matter of fact, the classification for the parameters (35,17,8), (56,11,2), and (91,10,1) had already been done, and in this paper we present the results for the parameters (36,15,6), (40,13,4), and (45,12,3). We also provide information about the order and the structure of the full automorphism groups of the constructed designs. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 301–312, 2006