Finite Nonabelian Simple Group Research Articles

Maximal orders of Abelian subgroups in finite simple groups

We bring out upper bounds for the orders of Abelian subgroups in finite simple groups. (For alternating and classical groups, these estimates are, or are nearly, exact.) Precisely, the following result, Theorem A, is proved. Let G be a non-Abelian finite simple group and G $$\tilde \ne $$ L2 (q), where q=pt for some prime number p. Suppose A is an Abelian subgroup of G. Then |A|3<|G|. Our proof is based on a classification of finite simple groups. As a consequence we obtain Theorem B, which states that a non-Abelian finite simple group G can be represented as ABA, where A and B are its Abelian subgroups, iff G≌ L2(2t) for some t ≥ 2; moreover, |A|-2t+1, |B|=2t, and A is cyclic and B an elementary 2-group.

On Representing Finite Lattices as Intervals in Subgroup Lattices of Finite Groups

Let Mnbe the lattice of length 2 withn≥1 atoms. It is an open problem to decide whether or not every such lattice (or indeed whether or not every finite lattice) can be represented as an interval in the subgroup lattice of some finite group. We complete the work of the second author, Lucchini, by reducing this problem to a series of questions concerning the finite non-abelian simple groups.

Generating triples of involutions for lie-type groups over a finite field of odd characteristic. II

We describe simple Lie-type groups of rank l≤3 over a finite field of odd characteristic, generated by three involutions of which two are commuting. Previously, the answers to the similar questions were given for alternating groups, for Lie-type groups over a finite field of characteristic 2, and for Lie-type groups of rank l≥4 over a finite field of odd characteristic. These furnish a description of finite simple non-Abelian groups, distinct from 26 sporadic groups, generated by three involutions two of which are commuting, thus giving a partial answer to question 7.30 posed by Mazurov in the Kourovka Notebook.

A note on powers in simple groups

In [7], the second author proved that there is an integer k such that every element of a finite non-abelian simple group S is a product of k commutators in S. The motivation for proving this result came from a model-theoretic question about simple groups. The proof depended on the classification of the finite simple groups, a theorem of Malle, Saxl and Weigel [5] which shows that in many finite simple classical groups S there is a real conjugacy class R such that S=R3∪{1}, and an ultraproduct argument. Here we shall use a similar combination of ideas to prove the following result.

On the (2,3)-Generation of Automorphism Groups of Free Groups

A group is said to be (2,3)-generated if it can be generated by an element of order 2 and an element of order 3. The class of (2,3)-generated groups seems to be rather extensive. From results of Schupp [7] and Mason and Pride [3], it is known that there are 2 ℵ 0 isomorphism classes of (2,3)-generated groups, and moreover that every countable group can be embedded in a (2,3)-generated group. All finite non-abelian simple groups are (2,3)-generated, with the exception of some groups of low rank in characteristics 2 and 3 (see Wilson [13] or Sanchini and Tamburini [6]). 1991 Mathematics Subject Classification 20E05, 20F28.

Order evaluation of products of subsets in finite groups and its applications. II

In this paper we give a new estimate of the cardinality of the product of subsets A B AB in a finite non-abelian simple group, where A A is normal and B B is arbitrary. This estimate improves the one given in J. Algebra 182 (1996), 577–603.

Infinite products of finite simple groups

We classify the sequences ⟨ S n ∣ n ∈ N ⟩ \langle S_{n} \mid n \in \mathbb {N} \rangle of finite simple nonabelian groups such that ∏ n S n \prod _{n} S_{n} has uncountable cofinality.

The finite simple groups with at most two fusion classes of every order

Elements a,b of a group G are said to be fused if a = bσ and to be inverse-fused if a =(b-1)σ for some σ ϵ Aut(G). The fusion class of a ϵ G is the set {aσ | σ ϵ Aut(G)}, and it is called a fusion class of order i if a has order iThis paper gives a complete classification of the finite nonabelian simple groups G for which either (i) or (ii) holds, where: (i) G has at most two fusion classes of order i for every i (23 examples); and (ii) any two elements of G of the same order are fused or inversenfused. The examples in case (ii) are: A5, A6,L2(7),L2(8), L3(4), Sz(8), M11 and M23An application is given concerning isomorphisms of Cay ley graphs.

A note on growth sequences of finite simple groups

The aim of this paper is to give a new precise formula for h(n, A), where A is a finite non-abelian simple group, h(n, A) is the maximum number such that Ah(n, A) can ke generated by n elements, and n ≥ 2. P. Hall gave a formula for h(n, A) in terms of the Möbius function of the subgroup lattice of A; the new formula involves a concept called cospread associated with that of spread as explained in Brenner and Wiegold (1975).

The Graph of the Truncated Icosahedron and the Last Letter of Galois

SO(3) it is the proper symmetry group of the icosahedron (and the dodecahedron). It is the unique finite subgroup of SO(3) which equals its own commutator subgroup—in fact, it is the unique non-abelian simple finite subgroup of SO(3). It can be thought of as the beginning or smallest member of the family of non-abelian finite simple groups. From the point of view of the McKay correspondence it (or more exactly its “double cover”) corresponds to the exceptional Lie group E8. A group isomorphic to A5 will be referred to as an icosahedral group and any structure admitting such a group as a symmetry group is said to have icosahedral symmetry. Given the unique role of the icosahedral group in group theory, any natural structure having icosahedral symmetry surely deserves special attention. In this paper we will be concerned with one such structure—a structure which seems to be appearing with increasing frequency in the scientific literature. Prior to the discovery of the Fullerenes, around ten years ago, the only known form of pure solid carbon was graphite and diamonds. These two forms are crystalline materials where the bonds between the carbon atoms exhibit hexagonal and tetrahedral structures, respectively. In neither of these two substances, however, are there isolated molecules of pure carbon. On the other hand, in

(2,3,t)-Generations for the janko group j3

A group G is (l,m,n)-generated if it is a quotient group of the triangle group T(l,m,n) = (x,y,z|x l= y m= z n= xyz= 1). In [8] the problem is posed to find all possible (l,m,n)-generations for the non-abelian finite simple groups. In this paper we partially answer this question for the Janko group J 3. We find all (2, 3, t)-generations as well as (2, 2,2,p)-generations, p a prime, for J 3

Residual properties of free groups, III

In this paper we want to prove the following theorem: Letx be an infinite set of non-abelian finite simple groups. Then the free groupF2 on 2 generators is residuallyx. This answers a question first posed by W. Magnus and later by A. Lubotzky [9], Yu. Gorchakov and V. Levchuk [4].

Residual properties of free groups, II

In this paper we prove that if X is an infinite class of flnite simple classical groups, then F2, the free group of rank 2, is residually X. This solves a special case of a question of W.Magnus. He conjectures that F2 is residually X for any infinite class X of finite non-abelian simple groups.

Character degrees of simple groups

Gluck [2] and others have investigated the relationship between p(G), the set of primes dividing the degrees of the irreducible complex characters of G, and o(G), the greatest number of primes dividing the degree of a single irreducible character of G. when G is a finite solvable group. For such groups Gluck [2] has shown that 1 p(G)/ ,< (g(G))‘+ 100(G). In this paper we examine the relationship between these two quantities for G a finite nonabelian simple group. In order to state our results we need to introduce some notation. If G is a finite group and S is a subset of Irr(Gj, the set of irreducible characters of G, we will say S is a covering set of G if for every prime divisor p of / G 1 there is a character x in S such that p divides x( 1). If G has a covering set we will define the covering number of G, which we will denote by en(G), as the least number of elements in a covering set of G. Work of Michler [6, Theorem 3.31 has as a consequence that if G is a nonabelian simple group then G has a covering set; in other words, p(G) = n(G), where n(G) is the set of prime divisors of the order of G. For such groups then en(G) ,( 1 r(G)1 However, more is true.

Diameter, Covering Index, Covering Radius and Eigenvalues

Fan Chung has recently derived an upper bound on the diameter of a regular graph as a function of the second largest eigenvalue in absolute value. We generalize this bound to the case of bipartite biregular graphs, and regular directed graphs.We also observe the connection with the primitivity exponent of the adjacency matrix. This applies directly to the covering number of Finite Non Abelian Simple Groups (FINASIG). We generalize this latter problem to primitive association schemes, such as the conjugacy scheme of Paige's simple loop.By noticing that the covering radius of a linear code is the diameter of a Cayley graph on the cosets, we derive an upper bound on the covering radius of a code as a function of the scattering of the weights of the dual code. When the code has even weights, we obtain a bound on the covering radius as a function of the dual distance dl which is tighter, for d⊥ large enough, than the recent bounds of Tietäväinen.

Small-diameter Cayley Graphs for Finite Simple Groups

Let S be a subset generating a finite group G . The corresponding Cayley graph G ( G , S ) has the elements of G as vertices and the pairs { g , sg }, g ∈ G , s ∈ S , as edges. The diameter of G ( G , S ) is the smallest integer d such that every element of G can be expressed as a word of length ⩿ d using elements from S ∪ S −1 . A simple count of words shows that d ⩾ log 2 ❘s❘ (❘ G ❘). We prove that there is a constant C such that every nonabelian finite simple group has a set S of at most 7 generators for which the diameter of G ( G , S ) is at most C log ❘ G ❘.

On the diameter of cayley graphs of the symmetric group

Let G be either the symmetric or the alternating group of degree n. We prove that, given any set S of generators of G, every member of G can be represented as a word in S ∪ S −1 of length not exceeding exp((n lnn) 1 2 (1 + o(1))) . We conjecture that the true bound is n const and we state an analogous conjecture for all nonabelian finite simple groups.

A classification of the maximal subgroups of the finite alternating and symmetric groups

Following the classification of finite simple groups, one of the major problems in finite group theory today is the determination of the maximal subgroups of the almost simple groups-that is, of groups X such that X0 u X< Aut X, for some finite non-abelian simple group X0. The problem has been solved for most sporadic groups and groups of Lie type of low rank (see [ 19, Sects. 3, 43 for discussion and references). This paper is a contribution to the case where X0 is an alternating group A, (so that for n # 6, X is A, or S,). The maximal subgroups of A, and S, are known for several classes of degrees n:

Almost-free groups in varieties with torsion

Let V be a variety of groups. A group G is said to be almost V -free if every subgroup of G that can be generated by fewer elements than the cardinality of G is contained in a V -free subgroup. It is shown in this paper that if V is a variety in which there is a finite non-abelian simple group, then there are for each positive integer n, 2 ℵ n almost V -free groups of cardinality ℵ n .

Simple groups with a small number of conjugacy classes

All finite non-Abelian simple groups whose number of conjugacy classes does not exceed 10 are computed (with application of a computer).