Finite Nonabelian Simple Group Research Articles

Presentations of finite simple groups: A quantitative approach

There is a constantC0C_0such that all nonabelian finite simple groups of ranknnoverFq\mathbb {F}_q, with the possible exception of the Ree groups2G2(32e+1)^2G_2(3^{2e+1}), have presentations with at mostC0C_0generators and relations and total length at mostC0(log⁡n+log⁡q)C_0(\log n +\log q). As a corollary, we deduce a conjecture of Holt: there is a constantCCsuch thatdim⁡H2(G,M)≤Cdim⁡M\dim H^2(G,M) \leq C\dim Mfor every finite simple groupGG, every primeppand every irreducibleFpG{\mathbb F}_p G-moduleMM.

A New Characterization of A 10 by its Noncommuting Graph

Let G be a nonabelian group and associate a noncommuting graph ∇(G) with G as follows: The vertex set of ∇(G) is G\\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, Professor J. G. Thompson gave the following conjecture. Thompson's Conjecture. If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ≅ M, where N(G):={n ∈ ℕ | G has a conjugacy class of size n}. In 2006, A. Abdollahi, S. Akbari, and H. R. Maimani put forward a conjecture (AAM's conjecture) in Abdollahi et al. (2006) as follows. AAM's Conjecture. Let M be a finite nonabelian simple group and G a group such that ∇(G) ≅ ∇ (M). Then G ≅ M. In this short article we prove that if G is a finite group with ∇(G) ≅ ∇ (A 10), then G ≅ A 10, where A 10 is the alternating group of degree 10.

Noncyclic Graph of a Group

We associate a graph Γ G to a nonlocally cyclic group G (called the noncyclic graph of G) as follows: take G\\ Cyc(G) as vertex set, where Cyc(G) = {x ∈ G| ⟨x, y⟩ is cyclic for all y ∈ G}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of Γ G is finite if and only if Γ G has no infinite clique. We prove that if G is a finite nilpotent group and H is a group with Γ G ≅ Γ H and |Cyc(G)| = |Cyc(H)| = 1, then H is a finite nilpotent group. We give some examples of groups G whose noncyclic graphs are “unique”, i.e., if Γ G ≅ Γ H for some group H, then G ≅ H. In view of these examples, we conjecture that every finite nonabelian simple group has a unique noncyclic graph. Also we give some examples of finite noncyclic groups G with the property that if Γ G ≅ Γ H for some group H, then |G| = |H|. These suggest the question whether the latter property holds for all finite noncyclic groups.

Products of Subgroups Which Are Subgroups

We prove conditions for a product of distinct subgroups of an arbitrary group G to be a subgroup of G. In particular, the normal closure of any A ≤ G is equal to the product of some distinct conjugates of A. As an application of the later result we derive constraints on the size of a nontrivial conjugacy class of a finite non-Abelian simple group.

Covering Theorem for Finite Nonabelian Simple Groups

In this paper, we show that for the alternating group An, the class C of n- cycle, CC covers An for n when n = 4k + 1 > 5 and odd. This class splits into two classes of An denoted by C and C/, CC= C/C/ was found.

On the growth sequences of PSp(2m,q)

The aim of this paper is to give a lower bound for h(2,PSp(2m,q)), for all 2 ≤ m ≤ 5, m ≥ 10 and q ≥ 2, where h(2,G) is the maximum number such that G h(2,G) can be generated by 2 elements. Furthermore, we consider a problem which was conjectured by J.Wiegold and the first author in 1996, which says that h(2,G) 2 > |G| for all finite non-abelian simple groups. We confirm the conjecture for the projective symplectic simple groups PSp(2m,q) at the end.

Localization and finite simple groups

A homomorphism from a groupH to a groupG is a localization if and only if it induces a bijection between Hom(G, G) and Hom(H, G). In this paper we study the equivalence relation that localization induces on the family of finite non-abelian simple groups.

Unfused Involutions in Finite Groups: An Addendum

If p is a prime, r is an integer and p r > 2, there exist infinitely many finite simple non-Abelian groups containing an element x of order p r such that if P is a Sylow p-subgroup of G containing x, x G ∩ P = x P .

Finite simple groups with complemented maximal subgroups

We study Problem 8.31 in [1] of the description of finite weakly factorizable groups, i.e., the groups whose every proper subgroup is complemented in a larger subgroup. By Lemma 1 of [2], the subdirect product of a completely factorizable group (such groups were studied by Ph. Hall and N. V. Baeva (Chernikova)) and a weakly factorizable group is a weakly factorizable group. In connection with a remark in [2], we observe that a dihedral 2-group is always weakly factorizable but, in general, it cannot even be obtained from the groups of prime order by repeated application of the lemma. Theorem 1, basing on the available maximal factorizations, shows that there exist exactly three finite simple nonabelian groups with complemented maximal subgroups. Theorem 2 confirms the conjecture of [2] on uniqueness of a finite simple nonabelian group with the property of weak factorizability. Earlier Theorems 1 and 2 were proven in special cases by A. G. Likharev.

PROBABILISTIC GENERATION OF WREATH PRODUCTS OF NON-ABELIAN FINITE SIMPLE GROUPS, II

We show that the probability of generating an iterated wreath product of non-abelian finite simple groups converges to 1 as the order of the first simple group tends to infinity provided the wreath products are constructed with transitive and faithful actions. This has the consequence that the profinite group which is the inverse limit of these iterated wreath products is positively finitely generated.

NX-complementary generations of the Fischer group Fi 23

Let G be a finite group and nX a conjugacy class of elements of order n in G. G is called nX complementary generated if, for every x 2 G { 1}, there is a y 2 nX such that G = hx,yi. In (27) the question of finding all positive integers n such that a given non-abelian finite simple group G is nX complementary generated was posed. In this paper we answer this question for the sporadic group Fi23. In fact, we prove that the Fischer group Fi23 is nX-complementary gen- erated if and only if n> 12 or n 2{ 7,8,10,11} or nX 2{ 6N,6O,9D,9E,

On finite simple groups acting on integer and mod 2 homology 3-spheres

We prove that the only finite non-abelian simple groups G which possibly admit an action on a Z 2 -homology 3-sphere are the linear fractional groups PSL ( 2 , q ) , for an odd prime power q (and the dodecahedral group A 5 ≅ PSL ( 2 , 5 ) in the case of an integer homology 3-sphere), by showing that G has dihedral Sylow 2-subgroups and applying the Gorenstein–Walter classification of such groups. We also discuss the minimal dimension of a homology sphere on which a linear fractional group PSL ( 2 , q ) acts.

$(p,q,r)$-GENERATIONS OF THE SPORADIC GROUP $HN$

A finite group $G$ is called $(l,m,n)$-generated, if it is a quotient group of the triangle group $T(l,m,n) = \langle x,y,z | x^l = y^m = z^n = xyz = 1 \rangle$. In [16], the question of finding all triples $(p,q,r)$ such that non-abelian finite simple group $G$ is $(p,q,r)-$generated was posed. In this paper we partially answer this question for the sporadic group $HN$. In fact, we prove that the sporadic group $HN$ is $(p,q,r)-$generated if and only if $(p,q,r) \ne (2,3,5)$, where $p, q$ and $r$ are prime divisors of $|HN|$ and $p \lt q \lt r$.

5-Arc transitive cubic Cayley graphs on finite simple groups

In this paper, we determine all connected 5-arc transitive cubic Cayley graphs on the alternating group A 47 ; there are only two such graphs (up to isomorphism). By earlier work of the authors, these are the only two non-normal connected cubic arc-transitive Cayley graphs for finite nonabelian simple groups, and so this paper completes the classification of such non-normal Cayley graphs.

On finite simple and nonsolvable groups acting on homology 4-spheres

The only finite non-Abelian simple group acting on a homology 3-sphere—necessarily non-freely—is the dodecahedral group A 5 ≅ PSL ( 2 , 5 ) (in analogy, the only finite perfect group acting freely on a homology 3-sphere is the binary dodecahedral group A 5 ∗ ≅ SL ( 2 , 5 ) ). In the present paper we show that the only finite simple groups acting on a homology 4-sphere, and in particular on the 4-sphere, are the alternating or linear fractional groups A 5 ≅ PSL ( 2 , 5 ) and A 6 ≅ PSL ( 2 , 9 ) . From this we deduce a short list of groups which contains all finite nonsolvable groups admitting an action on a homology 4-sphere.

Permutation groups, simple groups, and sieve methods

We show that the number of integersn≤x which occur as indices of subgroups of nonabelian finite simple groups, excluding that ofA n−1 inA n , is ∼hx/logx, for some given constanth. This might be regarded as a noncommutative analogue of the Prime Number Theorem (which counts indicesn≤x of subgroups of abelian simple groups).We conclude that for most positive integersn, the only quasiprimitive permutation groups of degreen areS n andA n in their natural action. This extends a similar result for primitive permutation groups obtained by Cameron, Neumann and Teague in 1982.Our proof combines group-theoretic and number-theoretic methods. In particular, we use the classification of finite simple groups, and we also apply sieve methods to estimate the size of some interesting sets of primes.

An Adjacency Criterion for the Prime Graph of a Finite Simple Group

In the paper we give an exhaustive arithmetic criterion of adjacency in prime graph $GK(G)$ for every finite nonabelian simple group $G$. By using this criterion for all finite simple groups an independence set with the maximal number of vertices, an independence set containing 2 with the maximal number of vertices, and the orders of these independence sets are given. We assemble this information in the tables at the end of the paper. Several applications of obtained results for various problems of finite group theory are considered.

Finite Non-abelian Simple Groups Which Contain a Non-trivial Semipermutable Subgroup

A subgroup H of a group G is called semipermutable if it is permutable with every subgroup K of G with (|H|,|K|)=1, and s-semipermutable if it is permutable with every Sylow p-subgroup of G with (p,|H|)=1. In this paper, we classify all finite non-abelian simple groups which contain a non-trivial semipermutable (s-semipermutable) subgroup. As a corollary of our main result, we give a complete answer to an unsolved problem in group theory proposed by V.S. Monakhov in 1990.

An Elementary Approach to the Monster

is one of the most spectacular and mysterious mathematical achievements of the past fifty years. Unfortunately the various standard approaches to the Monster are notoriously difficult to learn. This paper presents a relatively elementary approach to the Monster. We describe a construction of the smallest non-Abelian finite simple group (of order 60), and show that it very closely parallels a construction of M x M. In section 2, before we begin our main discussion, we give a quick overview of the Monster both as a finite simple group and with respect to its most startling properties. This is done to allow those readers who have only a passing acquaintance with the Monster to get a better sense of its role in mathematics. Despite its brevity, this section has a vast scope, so we refer the reader to other sources for more comprehensive treatments (see [1], [2], or [6]). In section 3 we start with the tetrahedral graph and use Coxeter groups, along with certain additional relations, to obtain a group presentation for the finite simple group As. Then in section 4 we see that this method, when applied to different initial graphs, leads not only to the square M x M of the elusive Monster group, but to other finite simple groups as well.

On recognition of all finite nonabelian simple groups with orders having prime divisors at most 13

The spectrum of a group is the set of its element orders. We say that the problem of recognition by spectrum is solved for a finite group if we know the number of pairwise nonisomorphic finite groups with the same spectrum as the group under study. In this article the problem of recognition by spectrum is completely solved for every finite nonabelian simple group with orders having prime divisors at most 13.