Finite Nonabelian Simple Group Research Articles

Classifying Finite Simple Groups with Respect to the Number of Orbits Under the Action of the Automorphism Group#

Let ω(G) denote the number of orbits on the finite group G under the action of Aut(G). Using the classification of finite simple groups, we prove that for any positive integer n, there is only a finite number of (non-abelian) finite simple groups G satisfying ω(G) ≤ n. Then we classify all finite simple groups G such that ω(G) ≤ 17. The latter result was obtained by computational means, using the computer algebra system GAP.

Probabilistic Generation of Wreath Products of Non-abelian Finite Simple Groups#

We show that the probability of generating an iterated standard wreath product of non-abelian finite simple groups is positive and tends to 1 as the order of the first simple group tends to infinity. This has the consequence that the profinite group which is the inverse limit of these iterated wreath products is positively finitely generated. Information depending on the Classification of Finite Simple Groups is used throughout.

On cubic s-arc transitive Cayley graphs of finite simple groups

For a positive integer s, a graph Γ is called s- arc transitive if its full automorphism group Aut Γ acts transitively on the set of s-arcs of Γ. Given a group G and a subset S of G with S= S −1 and 1∉ S, let Γ=Cay( G, S) be the Cayley graph of G with respect to S and G R the set of right translations of G on G. Then G R forms a regular subgroup of Aut Γ. A Cayley graph Γ=Cay( G, S) is called normal if G R is normal in Aut Γ. In this paper we investigate connected cubic s-arc transitive Cayley graphs Γ of finite non-Abelian simple groups. Based on Li’s work (Ph.D. Thesis (1996)), we prove that either Γ is normal with s≤2 or G= A 47 with s=5 and Aut Γ ≅ A 48. Further, a connected 5-arc transitive cubic Cayley graph of A 47 is constructed.

Cyclic branched coverings and homology 3-spheres with large group actions

We show that, if the covering involution of a $3$-manifold $M$ occurring as the $2$-fold branched covering of a knot in the $3$-sphere is contained in a finite nonabelian simple group $G$ of diffeomorphisms of $M$, then $M$ is a homology $3$-sphere and $

ON THE COMMUTATOR WIDTH OF PERFECT GROUPS

This paper proves bounds for the commutator width of a wreath product of two groups. As a corollary, it is shown that the commutator width of finite perfect linear groups of dimension 15 is unbounded. It follows that the covering number of these groups is unbounded. On the other hand, the commutator width of iterated wreath products of nonabelian finite simple groups is bounded by an absolute constant.

(p,q,r)-Generations and nX-Complementary Generations of the Thompson Group Th

A group G is said to be (l,m,n)-generated if it is a quotient group of the triangle group T(l,m,n)=〈x,y,z|xl=ym=zn=xyz=1〉. In 1993 J. Moori posed the question of finding all triples (l,m,n) such that a given non-abelian finite simple group is (l,m,n)-generated. In this paper we partially answer this question for the Thompson group Th. In fact we study (p,q,r)-generation, where p, q and r are distinct primes, and nX-complementary generations of the Thompson group Th.

Generating pairs for the sporadic groupRu

A finite group G is called (l,m,n)-generated, if it is a quotient group of the triangle group T(l,m,n) = (x,y, z|xl = ym = zn = xyz = 1). In [19], the question of finding all triples (l,m,n) such that non-abelian finite simple groups are (l,m,n)-generated was posed. In this paper we partially answer this question for the sporadic group Ru. In fact, we prove that if p,q and r are prime divisors of |Ru|, where p < q r and (p,q) ≠ (2, 3), then Ru is (p,q,r)-generated.

Smith equivalence and finite Oliver groups with Laitinen number 0 or 1

In 1960, Paul A. Smith asked the following question. If a finite group G acts smoothly on a sphere with exactly two fixed points, is it true that the tangent G-modules at the two points are always isomorphic? We focus on the case G is an Oliver group and we present a classification of finite Oliver groups G with Laitinen number a_G = 0 or 1. Then we show that the Smith Isomorphism Question has a negative answer and a_G > 1 for any finite Oliver group G of odd order, and for any finite Oliver group G with a cyclic quotient of order pq for two distinct odd primes p and q. We also show that with just one unknown case, this question has a negative answer for any finite nonsolvable gap group G with a_G > 1. Moreover, we deduce that for a finite nonabelian simple group G, the answer to the Smith Isomorphism Question is affirmative if and only if a_G = 0 or 1.

Dominions in varieties generated by simple groups

We determine the dominion (in the sense of Isbell) of a subgroup H of a finite nonabelian simple group S in Var(S). We also obtain necessary and suficient conditions for H to be epimorphically embedded in S, and derive several examples, which generalize an example of B. H. Neumann. Finally, we extend the results to a variety generated by a family of finite nonabelian simple groups.

Generating pairs for the held groupHe

A group G is said to be (l, m, n)-generated if it is a quotient group of the triangle group T(p,q,r) = 〈x,y,z|xp = yq = zr = xyz = 1〉. In [15], the question of finding all triples (l, m, n) such that non-abelian finite simple groups are (l, m, n)-generated was posed.In this paper we partially answer this question for the sporadic group He. We continue the study of (p, q, r)-generations of the sporadic simple groups, where p, q, r are distinct primes. The problem is resolved for the Held group He.

On 1-Arc-regular Graphs

A graph is 1-arc-regular if its full automorphism group acts regularly on the set of its arcs. In this paper, we investigate 1-arc-regular graphs of prime valency, especially of valency 3. First, we prove that if G is a soluble group then a (G, 1)-arc-regular graph must be a Cayley graph of a subgroup of G. Next we consider trivalent Cayley graphs of a finite nonabelian simple group and obtain a sufficient condition under which one can guarantee that Cay(G, S) is 1-arc-regular. Finally, as an application of the result, we construct two infinite families of 1-arc-regular trivalent Cayley graphs with insoluble automorphism groups and, in particular, one of the families is not a Cayley graph.

Anisotropic groups of type Anand the commuting graph of finite simple groups

In this paper we make a contribution to the MargulisPlatonov conjecture, which describes the normal subgroup structure of algebraic groups over number fields. We establish the conjecture for inner forms of anisotropic groups of type An. We obtain information on the commuting graph of nonabelian finite simple groups, and consequently, using the paper by Segev, 1999, we obtain results on the normal structure and quotient groups of the multiplicative group of a division algebra.

On finite simple groups acting on homology 3-spheres

We show that the only nonabelian finite simple group which admits smooth actions on a homology 3-sphere is the dodecahedral or alternating group A5.

The finite images of finitely generated groups

Given any sequence of non-abelian finite simple primitive permutation groups Sn, we construct a finitely generated group G whose profinite completion is the infinite permutational wreath product … Sn ≀ Sn−1 ≀ … ≀ S0. It follows that the upper composition factors of G are exactly the groups Sn. By suitably choosing the sequence Sn we can arrange that G has any one of a continuous range of slow, non-polynomial subgroup growth types. We also construct a 61-generator perfect group that has every non-abelian finite simple group as a quotient. 2000 Mathematics Subject Classification: 20E07, 20E08, 20E18, 20E32.

Locally primitive Cayley graphs of finite simple groups

A graph Γ is said to be G-locally primitive , where G is a subgroup of automorphisms of Γ , if the stabiliser G α of a vertex α acts primitively on the set Γ ( α ) of vertices of Γ adjacent to α . For a finite non-abelian simple group L and a Cayley subset S of L , suppose that L G ≤Aut( L ), and the Cayley graph Γ =Cay ( L,S ) is G -locally primitive. In this paper we prove that L is a simple group of Lie type, and either the valency of Γ is an add prine divisor of |Out( L )|, or L=P Ω 8 + ( q ) and Γ has valency 4. In either cases, it is proved that the full automorphism group of Γ is also almost simple with the same socle L .

Large nilpotent subgroups of finite simple groups

Orders and the structure of large nilpotent subgroups in all finite simple groups are determined. In particular, it is proved that if G is a finite simple non-Abelian group, and N is some of its nilpotent subgroups, then |N|2<|G|.

Finite Simple Groups of Bounded Subgroup Chain Length

Given a finite group G, we denote by l(G) the length of the longest chain of subgroups of G. We study whether certain sets of non-isomorphic finite simple groups S with bounded l(S) are finite or infinite. We prove, in particular, that there exists an infinite number of non-isomorphic non-abelian finite simple groups S with l(S) bounded.

Non-abelian Sylow subgroups of finite groups of even order

We have been able to prove that every nonabelian Sylow subgroup of a finite group of even order contains a nontrivial element which commutes with an involution. The proof depends upon the consequences of the classification of finite simple groups. The purpose of this note is to announce [4]: Main Theorem. Every nonabelian Sylow subgroup of a finite group of even order contains a nontrivial element which commutes with an involution. Let G be a finite group and Γ(G) the prime graph of G. Γ(G) is the graph such that the vertex set is the set of prime divisors of |G|, and two distinct vertices p and r are joined by an edge if and only if there exists an element of order pr in G. Let n(Γ(G)) be the number of connected components of Γ(G) and dG(p, r) the distance between two vertices p and r of Γ(G). It has been proved that n(Γ(G)) ≤ 6 in [12], [10], [11], [9]. Theorem 1. Let G be a finite group of even order and p a prime divisor of |G|. If dG(2, p) ≥ 2, then a Sylow p-subgroup of G is abelian. Theorem 1 is a restatement of the Main Theorem in terms of the prime graph Γ(G). Corollary 1. Let G be a finite group of even order and p a prime divisor of |G|. If ∆ is a connected component of Γ(G)−{p} not containing 2, then a Sylow r-subgroup of G is abelian for any r ∈ ∆. There is a certain relation between a subgraph Γ(G) − {p} of Γ(G) and Brauer characters of p-modular representations of G (see [3]). Theorem 2. Let G be a finite nonabelian simple group and p an odd prime divisor of |G|. Then dG(2, p) = 1 or 2 provided dG(2, p) < ∞. The significance of the prime graphs of finite groups can be found in [1], [3], [5], [6], [7], [8], [14], [15]. We apply the classification of finite simple groups (see [1], [2], [4], [7], [10], [13]). It has been proved that a minimal counterexample to Theorem 1 Received by the editors October 20, 1997. 1991 Mathematics Subject Classification. Primary 20D05, 20D06, 20D20.

The characterization of finite simple groups with no elements of order six by their element orders

We show that apart from A6 all non-abelian finite simple groups with no elements of order 6 are characterized by their element orders.

Putting the squeeze on the Noether gap - The case of the alternating groups $A_n$

Let \(\rho : G \hookrightarrow{\rm GL}(n, \mathbb{F})\) be a faithful representation of the finite group G over the field \(\mathbb{F}\). In 1916 E. Noether proved that for \(\mathbb{F}\) of characteristic zero the ring of invariants \(\mathbb{F}[V]^G\) is generated as an algebra by the invariant polynomials of degree at most \(|G|\). This upper bound for the degrees of the generators in a minimal generating set is referred to as Noether's bound. The result has been generalized to the case where the characteristic of \(\mathbb{F}\) is greater than the order of G, or when the characteristic of \(\mathbb{F}\) is prime to the order of G, and the group G is solvable. In this note we prove that if Noether's bound fails when the order of the group is prime to the characteristic of \(\mathbb{F}\), referred to as the nonmodular case, then it fails for a finite nonabelian simple group. We then show how yet another reworking of Noether's argument leads to a proof that Noether's bound holds in the nonmodular case for the alternating groups.