On finite groups isospectral to simple linear and unitary groups

Let L be a simple linear or unitary group of dimension larger than 3 over a finite field of characteristic p. We deal with the class of finite groups isospectral to L. It is known that a group of this class has a unique nonabelian composition factor. We prove that if L ≠ U ₄(2), U ₅(2) then this factor is isomorphic to either L or a group of Lie type over a field of characteristic different from p.     

Invariants of modular groups

For a faithful linear representation of a finite group G over a field of characteristic p, we study the ring of invariants. We especially study the polynomial and Cohen–Macaulay properties of the invariant ring. We first show that certain quotient rings of the invariant ring are polynomial rings by which we prove that the Hilbert ideal conjecture is true for a class of groups. In particular, we prove that the conjecture is true for vector invariant rings of Abelian reflection p-groups. Then we study the relationships between the invariant ring of G and that of a subgroup of G. Finally, we study the invariant rings of affine groups and show that, over a finite field, if an affine group contains all translations then the invariant ring is isomorphic to the invariant ring of a linear group.     

Characterization of simple symplectic groups of degree 4 over locally finite fields of characteristic 2 in the class of periodic groups

Suppose that each finite subgroup of even order of a periodic group containing an element of order 2 lies in a subgroup isomorphic to a simple symplectic group of degree 4 over some finite field of characteristic 2. We prove that in that case the group is isomorphic to a simple symplectic group S ₄(Q) over some locally finite field Q of characteristic 2.     

Deciding finiteness of matrix groups in positive characteristic

We present a new algorithm to decide finiteness of matrix groups defined over a field of positive characteristic. Together with previous work for groups in zero characteristic, this provides the first complete solution of the finiteness problem for finitely generated matrix groups over a field. We also give an algorithm to compute the order of a finite matrix group over a function field of positive characteristic by constructing an isomorphic copy of the group over a finite field. Our implementations of these algorithms are publicly available in Magma.     

On the structure of compact torsion groups

It is shown that every profinite torsion group has a finite series of closed characteristic subgroups in which each factor either is a pro-p-group for some primep or is isomorphic to a Cartesian product of isomorphic finite simple groups.     

Conjugate biprimitive finite groups saturated with finite simple subgroupsU 3(2 n )

A group G is saturated with groups of the set X if every finite subgroup K≤G is embedded in G into a subgroup L isomorphic to some group of X. We study periodic conjugate biprimitive finite groups saturated with groups in the set {U₃(2ⁿ)}. It is proved that every such group is isomorphic to a simple group U₃(Q) over a locally finite field Q of characteristic 2. Supported by the RF State Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 5, pp. 606–615, September–October, 1998.     

Formal groups and zeta-functions of elliptic curves

The author shows that the isomorphism class of a formal group overZ/pZ (resp. overZ _p ) of finite height (resp. having reduction modp of finite height) is determined by its characteristic polynomial. It is then proved that the formal groups associated to a large class of Dirichlet series with integer coefficients are defined overZ.Finally, these results are used to extend a theorem of Honda (Osaka J. Math.5, 199–213 (1968), Theorem 5) to include the case of supersingular reduction at the primes 2 and 3. LetE be an elliptic curve defined overQ, andF(x, y) be a formal minimal model forE. LetG(x, y) be the formal group associated to the globalL-seriesL(E, s) ofE overQ. Honda's theorem now becomes:G(x, y) is defined over Z and is isomorphic over Z to F(x, y).     

Difference algebraic subgroups of commutative algebraic groups over finite fields

We study the question of which torsion subgroups of commutative algebraic groups over finite fields are contained in modular difference algebraic groups for some choice of a field automorphism. We show that if G is a simple commutative algebraic group over a finite field of characteristic p, ? is a prime different from p, and for some difference closed field (?, σ) the ?-primary torsion of G(?) is contained in a modular group definable in (?, σ), then it is contained in a group of the form {x∈G(?) :σ(x) =[a](x) } with a∈ℕ\p ^ℕ. We show that no such modular group can be found for many G of interest. In the cases that such equations may be found, we recover an effective version of a theorem of Boxall. Received: 28 May 1998 / Revised version: 20 December 1998     

On genus and embeddings of torsion-free nilpotent groups of class two

Summary We study embeddings between torsion-free nilpotent groups having isomorphic localizations. Firstly, we show that for finitely generated torsion-free nilpotent groups of nilpotency class 2, the property of having isomorphicP-localizations (whereP denotes any set of primes) is equivalent to the existence of mutual embeddings of finite index not divisible by any prime inP. We then focus on a certain family Γ of nilpotent groups whose Mislin genera can be identified with quotient sets of ideal class groups in quadratic fields. We show that the multiplication of equivalence classes of groups in Γ induced by the ideal class group structure can be described by means of certain pull-back diagrams reflecting the existence of enough embeddings between members of each Mislin genus. In this sense, the family Γ resembles the family N₀ of infinite, finitely generated nilpotent groups with finite commutator subgroup. We also show that, in further analogy with N₀, two groups in Γ with isomorphic localizations at every prime have isomorphic localizations at every finite set of primes. We supply counterexamples showing that this is not true in general, neither for finitely generated torsion-free nilpotent groups of class 2 nor for torsion-free abelian groups of finite rank. Supported by DGICYT grant PB94-0725 This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag.     

Automorphism groups of graphs with forbidden subgraphs

For a graph Ф letF(Ф) be the class of finite graphs which do not contain an induced subgraph isomorphic to Ф. We show that whenever Ф is not isomorphic to a path on at most 4 vertices or to the complement of such a graph then for every finite groupG there exists a graph ГєF(Ф) such thatG is isomorphic to the automorphism group of Г. For all paths д on at most 4 vertices we determine the class of all automorphism groups of members ofF(д).     

Basic Hopf algebras and quantum groups

This paper investigates the structure of basic finite dimensional Hopf algebras H over an algebraically closed field k. The algebra H is basic provided H modulo its Jacobson radical is a product of the field k. In this case H is isomorphic to a path algebra given by a finite quiver with relations. Necessary conditions on the quiver and on the coalgebra structure are found. In particular, it is shown that only the quivers given in terms of a finite group G and sequence of elements of G in the following way can occur. The quiver has vertices and arrows , where the set is closed under conjugation with elements in G and for each g in G, the sequences W and are the same up to a permutation. We show how is a kG-bimodule and study properties of the left and right actions of G on the path algebra. Furthermore, it is shown that the conditions we find can be used to give the path algebras themselves a Hopf algebra structure (for an arbitrary field k). The results are also translated into the language of coverings. Finally, a new class of finite dimensional basic Hopf algebras are constructed over a not necessarily algebraically closed field, most of which are quantum groups. The construction is not characteristic free. All the quivers , where the elements of W generates an abelian subgroup of G, are shown to occur for finite dimensional Hopf algebras. The existence of such algebras is shown by explicit construction. For closely related results of Cibils and Rosso see [Ci-R]. Received August 15, 1994; in final form May 16, 1997     

On group rings of linear groups

Let H be a finitely generated group of matrices over a field F of characteristic zero. We consider the group ring KH of H over an arbitrary field K whose characteristic is either zero or greater than some number $N=N(H)$. We prove that KH is isomorphic to a subring of a ring S which is a crossed product of a division ring Δ with a finite group. Hence KH is isomorphic to a subring of a matrix ring over a skew field.     

Conjugate biprimitive finite groups saturated with finite simple subgroups

A group G is saturated with groups of the set X if every finite subgroup K≤G is embedded in G into a subgroup L isomorphic to some group of X. We study periodic biprimitive finite groups saturated with groups of the sets {L₂(pⁿ)}, {Re(3²ⁿ⁺¹)}, and {Sz(2²ⁿ⁺¹)}. It is proved thai such groups are all isomorphic to {L₂(P)}, {Re(Q)}, or {Sr(Q)} over locally finite fields. Supported by the RF State Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 2, pp. 224–245, March–April, 1998.     

Finite Flat Group Schemes over Local Artin Rings

Let R be a local Artin ring with maximal ideal m and residue class field of characteristic p > 0. We show that every finite flat group scheme over R is annihilated by its rank, whenever m ^p = pm = 0. This implies that any finite flat group scheme over an Artin ring the square of whose maximal ideal is zero, is annihilated by its rank.     

Direct summands of class groups

Let G be a finite abelian group, and F a global field of characteristic prime to the order of G. Then there exists a finite extension of F whose class group has a direct summand isomorphic to G.     

Periodic groups saturated by finite simple groups <Emphasis Type="Italic">U</Emphasis><Subscript>3</Subscript>(2<Superscript>m</Superscript>)

Let {ie166-01} be a set of finite groups. A group G is said to be saturated by the groups in {ie166-02} if every finite subgroup of G is contained in a subgroup isomorphic to a member of {ie166-03}. It is proved that a periodic group G saturated by groups in a set {U₃(2^m) | m = 1, 2, …} is isomorphic to U₃(Q) for some locally finite field Q of characteristic 2; in particular, G is locally finite. __________ Translated from Algebra i Logika, Vol. 47, No. 3, pp. 288–306, May–June, 2008.     

Characterization of Locally Finite Simple Groups of the Type <Superscript>3</Superscript><Emphasis Type="Italic">D</Emphasis><Subscript>4</Subscript> Over Fields of Odd Characteristic in the Class of Periodic Groups

We prove that a periodic group is locally finite, given that each finite subgroup of the group lies in a subgroup isomorphic to a finite simple group of Lie type ³D₄ over a field of odd characteristic.     

Ultraproducts and Chevalley groups

Given a simple non-trivial finite-dimensional Lie algebra L, fields and Chevalley groups , we first prove that is isomorphic to . Then we consider the case of Chevalley groups of twisted type . We obtain a result analogous to the previous one. Given perfect fields having the property that any element is either a square or the opposite of a square and Chevalley groups , then is isomorphic to . We apply our results to prove the decidability of the set of sentences true in almost all finite groups of the form L(K) where K is a finite field and L a fixed untwisted Chevalley type. Received: 19 November 1993 / Revised version: 15 November 1995     

Transfer and Tate’s theorem

Let H be a subgroup of a finite group G, and assume that p is a prime that does not divide |G : H|. In favorable circumstances, one can use transfer theory to deduce that the largest abelian p-groups that occur as factor groups of G and of H are isomorphic. When this happens, Tate’s theorem guarantees that the largest not-necessarily-abelian p-groups that occur as factor groups of G and H are isomorphic. Known proofs of Tate’s theorem involve cohomology or character theory, but in this paper, a new elementary proof is given. It is also shown that the largest abelian p-factor group of G is always isomorphic to a direct factor of the largest abelian p-factor group of H. Received: 17 June 2008     

On the Loewy length of the center of a block with elementary abelian defect groups

Let G be a finite group and k an algebraically closed field of characteristic p. In this paper we investigate the Loewy structure of centers of indecomposable group algebras kG, for groups G with a normal elementary abelian Sylow p-subgroup. Furthermore, we show a reduction result for the case that a normal abelian Sylow p-subgroup is acted upon by a subgroup of its automorphism group; this is fundamental in providing generic formulae for the Loewy lengths considered.