ON NONABELIAN TENSOR PRODUCT MODULO q OF GROUPS

The nonabelian tensor product modulo q of two crossed modules of groups is investigated, where q is a positive integer. It is obtained a six term exact sequence of groups connecting the nonabelian tensor product modulo q with algebraic K-functor K ₂ with Z _q coefficients for (noncommutative) local rings. The notion of q-homology groups of a group G with coefficients in a G-module A is introduced, some its properties and calculations are given. The relationship between q-homology groups and derived functors of tensor product modulo q is studied.     

Tightening Turyn’s bound for Hadamard difference sets

This work examines the existence of (4q ²,2q ²−q,q ²−q) difference sets, for q=p ^f , where p is a prime and f is a positive integer. Suppose that G is a group of order 4q ² which has a normal subgroup K of order q such that G/K ≅ C _q ×C ₂×C ₂, where C _q ,C ₂ are the cyclic groups of order q and 2 respectively. Under the assumption that p is greater than or equal to 5, this work shows that G does not admit (4q ²,2q ²−q,q ²−q) difference sets.     

A reduction theorem for the McKay conjecture

The McKay conjecture asserts that for every finite group G and every prime p, the number of irreducible characters of G having p’-degree is equal to the number of such characters of the normalizer of a Sylow p-subgroup of G. Although this has been confirmed for large numbers of groups, including, for example, all solvable groups and all symmetric groups, no general proof has yet been found. In this paper, we reduce the McKay conjecture to a question about simple groups. We give a list of conditions that we hope all simple groups will satisfy, and we show that the McKay conjecture will hold for a finite group G if every simple group involved in G satisfies these conditions. Also, we establish that our conditions are satisfied for the simple groups PSL₂(q) for all prime powers q≥4, and for the Suzuki groups Sz(q) and Ree groups R(q), where q=2 ^e or q=3 ^e respectively, and e>1 is odd. Since our conditions are also satisfied by the sporadic simple group J ₁, it follows that the McKay conjecture holds (for all primes p) for every finite group having an abelian Sylow 2-subgroup.     

Degree Graphs of Simple Linear and Unitary Groups

Let G be a finite group and let cd (G) be the set of irreducible character degrees of G. The degree graph Δ(G) is the graph whose set of vertices is the set of primes that divide degrees in cd (G), with an edge between p and q if pq divides a for some degree a ? cd (G). We determine the graph Δ(G) for the finite simple groups of types A _?(q) and ² A _? (q ²), that is, for the simple linear and unitary groups.     

The depth of the maximal subgroups of Ree groups

We determine the combinatorial and the ordinary depth of the maximal subgroups of the simple Ree groups ²G₂(q). As an application of these calculations, we determine the subdegrees of primitive actions of the groups ²G₂(q). We also improve an earlier estimate of Burness, Liebeck, and Shalev on the base size of” non-standard” primitive actions of ²G₂(q) on the coset space of maximal subgroups from 3 to 2.     

Quasirecognition by the set of element orders of the groups E 6( q ) and 2 E 6( q )

We prove that if L is one of the simple groups E ₆(q) and ² E ₆(q) and G is some finite group with the same spectrum as L, then the commutant of G/F(G) is isomorphic to L and the quotient G/G′ is a cyclic {2,3}-group. Original Russian Text Copyright ? 2007 Kondrat’ev A. S. The author was supported by the Russian Foundation for Basic Research (Grant 04-01-00463) and the RFBR-NSFC (Grant 05-01-39000). __________ Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 6, pp. 1250–1271, November–December, 2007.     

Quotients of absolute Galois groups which determine the entire Galois cohomology

For a prime power q = p ^d and a field F containing a root of unity of order q we show that the Galois cohomology ring H^*(G_F,\mathbbZ/q){H^*(G_F,\mathbb{Z}/q)} is determined by a quotient G_F^[3]{G_F^{[3]}} of the absolute Galois group G _F related to its descending q-central sequence. Conversely, we show that G_F^[3]{G_F^{[3]}} is determined by the lower cohomology of G _F . This is used to give new examples of pro-p groups which do not occur as absolute Galois groups of fields.     

Maximal cyclic subgroups and prime divisors in finite groups

Let G be a finite group with no chief factor simple of Lie type E ₈(q) and C a cyclic subgroup of largest order in G. It is shown that at most two primes in the open interval ([|C|/2], |C|) divide |G|.Received: 14 January 2005     

Building fences around the chromatic coefficients

Associated to each graph G is its chromatic polynomial f(G, t) and we associate to f(G, t) the sequence α (G) of the norms of its coefficients. A stringent partial ordering is established for such sequences. The main result is that for any graph G with q edges we have α (R_q) ≤ α (G) ≤ α (S_q), where R_q and S_q are specified graphs with q edges. This translates into a clearer view of allowable values and patterns in the chromatic coefficients. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 123–128, 1997     

The Multiplicity Free Permutation Representations of the Ree Groups 2 G 2(q), the Suzuki Groups 2 B 2(q), and Their Automorphism Groups

Abstract

Let G a simple group of type ² B ₂(q) or ² G ₂(q), where q is an odd power of 2 or 3, respectively. The main goal of this paper is to determine the multiplicity free permutation representations of G and A ≤ Aut(G) where A is a subgroup containing a copy of G. Let B be a Borel subgroup of G. If G = ² B ₂(q) we show that there is only one non-trivial multiplicity free permutation representation, namely the representation of G associated to the action on G/B. If G = ² G ₂(q) we show that there are exactly two such non-trivial representations, namely the representations of G associated to the action on G/B and the action on G/M, where M = UC with U the maximal unipotent subgroup of B and C the unique subgroup of index 2 in the maximal split torus of B. The multiplicity free permutation representations of A correspond to the actions on A/H where H is isomorphic to a subgroup containing B if G = ² B ₂(q), and containing M if G = ² G ₂(q). The problem of determining the multiplicity free representations of the finite simple groups is important, for example, in the classification of distance-transitive graphs.     

On G-Covering Subgroup Systems for Some Saturated Formations of Finite Groups

Let ? be a class of groups and G a finite group. We call a set Σ of subgroups of G a G-covering subgroup system for ? if G ∈ ? whenever Σ ? ?. For a non-identity subgroup H of G, we put Σ _H be some set of subgroups of G which contains at least one supplement in G of each maximal subgroup of H. Let p ≠ q be primes dividing |G|, P, and Q be non-identity a p-subgroup and a q-subgroup of G, respectively. We prove that Σ _P and Σ _P  ∪ Σ _Q are G-covering subgroup systems for many classes of finite groups.     

Verifying Huppert’s Conjecture for PSp<Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>) when <Emphasis Type="Italic">q</Emphasis> > 7

Let G denote a finite group and cd (G) the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd (G) = cd (H), then G ≅ H × A, where A is an abelian group. Huppert verified the conjecture for PSp₄(q) when q = 3, 4, 5, or 7. In this paper, we extend Huppert’s results and verify the conjecture for PSp₄(q) for all q. This demonstrates progress toward the goal of verifying the conjecture for all nonabelian simple groups of Lie type of rank two.     

The Second Cohomology of Simple SL 3-Modules

Let G be the simple, simply connected algebraic group SL ₃ defined over an algebraically closed field K of characteristic p > 0. In this article, we find H ²(G, V) for any irreducible G-module V. When p > 7, we also find H ²(G(q), V) for any irreducible G(q)-module V for the finite Chevalley groups G(q) = SL(3, q) where q is a power of p.     

Aspects of the conjugacy class structure of simple algebraic groups

Let G be an adjoint simple algebraic group over an algebraically closed field of characteristic p; let Φ be the root system of G, and take t∈ℕ. Lawther has proven that the dimension of the set G _[t]={g∈G:g ^t =1} depends only on Φ and t. In particular the value is independent of the characteristic p; this was observed for t small and prime by Liebeck. Since G _[t] is clearly a disjoint union of conjugacy classes the question arises as to whether a similar result holds if we replace G _[t] by one of those classes. This paper provides a partial answer to that question. A special case of what we have proven is the following. Take p,q to be distinct primes and G _p and G _q to be adjoint simple algebraic groups with the same root system and over algebraically closed fields of characteristic p and q respectively. If s∈G _p has order q then there exists an element u∈G _q such that o(u)=o(s) and dimu^G_q=dims^G_p\dim u^{G_{q}}=\dim s^{G_{p}} .     

Cyclic caps in PG(3,q)

This article investigates cyclic completek-caps in PG(3,q). Namely, the different types of completek-capsK in PG(3,q) stabilized by a cyclic projective groupG of orderk, acting regularly on the points ofK, are determined. We show that in PG(3,q),q even, the elliptic quadric is the only cyclic completek-cap. Forq odd, it is shown that besides the elliptic quadric, there also exist cyclick-caps containingk/2 points of two disjoint elliptic quadrics or two disjoint hyperbolic quadrics and that there exist cyclick-caps stabilized by a transitive cyclic groupG fixing precisely one point and one plane of PG(3,q). Concrete examples of such caps, found using AXIOM and CAYLEY, are presented.     

On a class of transitive permutation groups of prime degreep=4n+1

SupposeG is a nonsolvable transitive permutation group of prime degreep, such that |N _G v(P)|=p(p−1) for some Sylowp-subgroupP ofG. Letq be a generator of the subgroup ofN _G (P), fixing one letter (it is easy to show that this subgroup is cyclic). Assume thatG contains an elementj such thatj ⁻¹ qj=q ^(p+1)/2. We shall prove that for almost all primesp of the formp=4n+1, a group that satisfies the above conditions must be the symmetric group on a set withp elements.     

A quasivariety lattice of torsion-free soluble groups

Let L _q (qG) be a lattice of quasivarieties contained in a quasivariety generated by a group G. It is proved that if G is a torsion-free finitely generated group in AB\mathcal{AB} _pk , where p is a prime, p ≠ 2, and k ∈ N, which is a split extension of an Abelian group by a cyclic group, then the lattice L _q (qG) is a finite chain.     

Designs from the Group PSL2(q), q Even

Let q be a power of 2 greater than 2 and consider the group G = PSL₂(q). We choose the maximal subgroups of G isomorphic to the dihedral groups D_2(q+1) and D_2(q-1) and present the primitive action of G on the right cosets of these two subgroups. We will find the orbits of the point stabilizer in each case and in the case of D_2(q-1) we will prove there is an orbit Δ of the point stabilizer G_ω, such that Δ ≠ {ω } and whose orbiting under G gives a 1-design with the automorphism group isomorphic to the symmetric group      

Character Tables of Parabolic Subgroups of the Chevalley Groups of Type G 2

The main aim of this paper is to construct the character tables of the parabolic subgroups of the Chevalley groups G ₂(q), where q is a power of a prime p > 3.