Random and Deterministic Triangle Generation of Three-Dimensional Classical Groups II

Let p ₁, p ₂, p ₃ be primes. This is the second article in a series of three on the (p ₁, p ₂, p ₃)-generation of the finite projective special unitary and linear groups PSU₃(p ⁿ ), PSL₃(p ⁿ ), where we say a noncyclic group is (p ₁, p ₂, p ₃)-generated if it is a homomorphic image of the triangle group T _{p 1, p 2, p 3} . This paper is concerned with the case where p ₁ = 2 and p ₂ = p ₃. We determine for any prime p ₂ the prime powers p ⁿ such that PSU₃(p ⁿ ) (respectively, PSL₃(p ⁿ )) is a quotient of T = T _{2, p 2, p 2} . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU₃(p ⁿ )) (respectively, Hom(T, PSL₃(p ⁿ ))) is surjective as p ⁿ tends to infinity.     

On Regular Orbits of Elements of Classical Groups in Their Permutation Representations

Let H be a simple finite classical group not isomorphic to PSL(n, q) for any n, q. We prove that every cyclic subgroup of H has a regular orbit on every nontrivial permutation H-set.     

Parametrization of Triangle Groups as Subgroups of PSL(2, q)

In this paper we are interested in triangle groups (j, k, l) where j = 2 and k = 3. The groups (j, k, l) can be considered as factor groups of the modular group PSL(2, Z) which has the presentation x, y : x² = y³ = 1. Since PSL(2,q) is a factor group of G^k,l,m if -1 is a quadratic residue in the finite field F_q, it is therefore worthwhile to look at (j, k, l) groups as subgroups of PSL(2, q) or PGL(2, q). Specifically, we shall find a condition in form of a polynomial for the existence of groups (2, 3, k) as subgroups of PSL(2, q) or PGL(2, q).Mathematics Subject Classification: Primary 20F05 Secondary 20G40.     

On the Cohopficity of the Direct Product of Cohopfian Groups

Suppose both A and B are cohopfian groups. Then A × B is cohopfian if A is either extremely noncommutative and torsion free, or finite Abelian, or finite simple.     

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

Abstract

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.     

Hurwitz Generation of PSp 6(q)

We show that the symplectic groups PSp₆(q) are Hurwitz for all q = p ^m  ≥ 5, with p an odd prime. The result cannot be improved since, for q even and q = 3, it is known that PSp₆(q) is not Hurwitz. In particular, n = 6 turns out to be the smallest degree for which a family of classical simple groups of degree n, over 𝔽 _{p ^m} , contains Hurwitz groups for infinitely many values of m. This fact, for a given (possibly large) p, also follows from [9 Larsen , M. , Lubotzky , A. , Marion , C. ( 2014 ). Deformation theory and finite simple quotients of triangle groups I . J. Eur. Math. Soc. (JEMS) 16 ( 7 ): 1349 – 1375 .[Crossref], [Web of Science ®] , [Google Scholar]] and [10 Larsen , M. , Lubotzky , A. , Marion , C. ( 2014 ). Deformation theory and finite simple quotients of triangle groups II . Groups Geom. Dyn. 8 ( 3 ): 811 – 836 .[Crossref], [Web of Science ®] , [Google Scholar]].     

The Invariant Fields of the Sylow Groups of Classical Groups in the Natural Characteristic

Let X be any finite classical group defined over a finite field of characteristic p > 0. In this article, we determine the fields of rational invariants for the Sylow p-subgroups of X, acting on the natural module. In particular, we prove that these fields are generated by orbit products of variables and certain invariant polynomials which are images under Steenrod operations, applied to the respective invariant linear forms defining X.     

On Permutation Groups of Degree a Product of Two Prime-Powers

We determine finite simple groups which have a subgroup of index with exactly two distinct prime divisors. Then from this we derive a classification of primitive permutation groups of degree a product of two prime-powers.     

Generation and random generation: From simple groups to maximal subgroups

Let G   be a finite group and let d(G)$d(G)$ be the minimal number of generators for G  . It is well known that d(G)=2$d(G)=2$ for all (non-abelian) finite simple groups. We prove that d(H)?4$d(H)?4$ for any maximal subgroup H of a finite simple group, and that this bound is best possible.     

Symmetric Presentations for the Fischer Groups II: The Sporadic Groups

We give symmetric presentations for groups closely related to the three sporadic Fischer groups Fi₂₂, Fi₂₃ and Fi₂₄.Mathematics Subject Classiffications (2000). 20D08, 20D06, 20F05This research was supported by an EPSRC grant ref. GR/N27491/01     

OD-Characterization of Alternating and Symmetric Groups of Degree p+5

In this paper,it is proved that all the alternating groups A_(p+5) are ODcharacterizable and the symmetric groups S_(p+5) are 3-fold OD-characterizable,where p + 4 is a composite number and p + 6 is a prime and 5≠p∈π(1000!).     

Groups with exactly two supercharacter theories

In this paper, we classify those finite groups with exactly two supercharacter theories. We show that the solvable groups with two supercharacter theories are ?₃ and S₃. We also show that the only nonsolvable group with two supercharacter theories is Sp(6,2).     

On Finite Groups with Few Automorphism Orbits

Denote by ω(G) the number of orbits of the action of Aut(G) on the finite group G. We prove that if G is a finite nonsolvable group in which ω(G) ≤5, then G is isomorphic to one of the groups A₅, A₆, PSL(2, 7), or PSL(2, 8). We also consider the case when ω(G) = 6 and show that, if G is a nonsolvable finite group with ω(G) = 6, then either G ≈ PSL(3, 4) or there exists a characteristic elementary abelian 2-subgroup N of G such that G/N ≈ A₅.     

Schur Algebras of Classical Groups II

We study Schur algebras of classical groups over an algebraically closed field of characteristic different from 2. We prove that Schur algebras are generalized Schur algebras (in Donkin's sense) in types A, C, and D, while this does not hold in type B. Consequently Schur algebras of types A, C, and D are integral quasi-hereditary by Donkin [7 Donkin , S. ( 1986 ). Schur algebras and related algebras I . J. Algebra 104 : 310 – 328 . [Google Scholar], 9 Donkin , S. ( 1994 ). Schur algebras and related algebras III: integral representations . Math. Proc. Camb. Phil. Soc. 116 : 37 – 55 . [Google Scholar]]. By using the coalgebra approach we put Schur algebras of a fixed classical group into a certain inverse system. We find that the corresponding hyperalgebra is contained in the inverse limit as a subalgebra. Moreover in types A, C, and D, the surjections in the inverse systems are compatible with the integral quasi-hereditary structure of Schur algebras.     

Nearly s-Normality of Groups and Its Properties

We call a subgroup H of a group G nearly s-normal in G if there exists N ? G such that HN ? G and H ∩ N ≤ H _sG , where H _sG is the largest s-permutable subgroup of G contained in H. In this article, we obtain some results about the nearly s-normal subgroups and use them to characterize the structure of finite groups.     

A New Characterization of the Mathieu Groups and Alternating Groups with Degrees Primes

In the past thirty years, several kinds of quantitative characterizations of finite groups especially finite simple groups have been investigated by many mathematicians. Such as quantitative characterizations by group order and element orders, by element orders alone, by the set of sizes of conjugacy classes, by dimensions of irreducible characters, by the set of orders of maximal abelian subgroups and so on. Here the authors continue this topic in a new area tending to characterize finite simple groups with given orders by some special conjugacy class sizes, such as largest conjugacy class sizes, smallest conjugacy class sizes greater than 1 and so on.     

Random Lazy Random Walks on Arbitrary Finite Groups

This paper considers lazy random walks supported on a random subset of k elements of a finite group G with order n. If k=a log₂ n where a>1 is constant, then most such walks take no more than a multiple of log₂ n steps to get close to uniformly distributed on G. If k=log₂ n+f(n) where f(n) and f(n)/log₂ n0 as n, then most such walks take no more than a multiple of (log₂ n) ln(log₂ n) steps to get close to uniformly distributed. To get these results, this paper extends techniques of Erdös and Rényi and of Pak.     

Indecomposable Sylow 2-Subgroups of Simple Groups

Let S be a Sylow 2-subgroup of a finite simple group and let S=S₁×S₂××S_k be the direct product and each component S_i, i=1,2,...,k is indecomposable. In this article, we prove that each S_i is also a Sylow 2-subgroup of some simple group. Mathematics Subject Classifications (2000) 20E32, 20D20.