Semisimplicity of the algebra of indecomposable representations of a finite commutative semigroup

Finite semigroups S, having only a finite number of indecomposable matrix representations over a field K of characteristic are considered. It is proved that the algebra over the field F of K-representations of the semigroup S is semisimple if and only if for any maximal subgroup of the semigroup S the characteristic of the field F does not divide the index of its Sylow p-subgroup and is greater than the order of this Sylow subgroup.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 57, pp. 125–128, 1976.     

On semigroup presentations and efficiency

The purpose of this paper is twofold. First we determine some forms of the relations in a finite semigroup presentation with zero deficiency which does or does not define a group. Moreover, we conclude that a finite Rees matrix semigroup M [G; I, Λ; P] is efficient when G is efficient and the index sets I, Λ are finite.     

Hypercentral Series and Paired Intersections of Sylow Subgroups of Chevalley Groups

Let G(k) be the Chevalley group of normal type associated with a root system G = , or of twisted type G = ^m,m = 2,3, over a field K. Its root subgroups X_s, for all possible s G⁺, generate a maximal unipotent subgroup U = UG(k) if p = charK < 0, U is a Sylow p-subgroup of G(K). We examine G and K for which there exists a paired intersection U U⁹, g G(K), which is not conjugate in G(K) to a normal subgroup of U. If K is a finite field, this is equivalent to a condition that the normalizer of U U⁹ in G(K)has a p-multiple index. Put p() = max(r,r)/(s,s) | r,s . We prove a statement (Theorem 1) saying the following. Let G(K) be a Chevalley group of Lie rank greater than 1 over a finite field K of characteristic p and U be its Sylow p-subgroup equal to UG(K); also, either G = and p() is distinct from p and 1, or G(K) is a twisted group. Then G(K) contains a monomial element n such that the normalizer U of Uⁿ in G(K) has a p-multiple index. Let K be an associative commutative ring with unity and (K,J) be a congruence subgroup of the Chevalley group (K) modulo a nilpotent ideal J. We examine an hypercentral series 1 Z₁ Z₂ ... Z_c-1 of the group U(K) (K,J). Theorem 2 shows that under an extra restriction on the quotient (J_t : J) of ideals, central series are related via Z_i = T_c-iC, 1 i < c, where C is a subgroup of central diagonal elements. Such a connection exists, in particular, if K = Z_pm and J = (p^d), 1 d < m, d| m.     

Subgroups of a finite group whose algebra of invariants is a complete intersection

Let G be a finite subgroup of GL(V), where V is a finite-dimensional vector space over the field K and char KG. We show that if the algebra of invariants K(V)^G of the symmetric algebra of V is a complete intersection then K(V)^H is also a complete intersection for all subgroups H of G such that H={ Gv (v)=v for all v V^H}.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 63–67, 1982.     

Cancellation over affine varieties

It is proved that if X is a smooth affine curve over a field F of characteristic , then the group SK₁(X)/ SK₁(X) is isomorphic to a subgroup of the étale cohomology group H _et ³ (X, _e ² ) and if F is algebraically closed, then SK₁(X) is a uniquely divisible group. The following cancellation theorem is obtained from results about SK₁ for curves: If X is a normal affine variety of dimension n over a field F, and if char F > n and C.d._e(F)1 for any prime >/n then any stably trivial vector bundle of rank n over X is trivial.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 187–195, 1982.     

On $$\mathfrak {F}_{hq}$$-supplemented subgroups of a finite group

A subgroup H of a finite group G is quasinormal in G if it permutes with every subgroup of G. A subgroup H of a finite group G is \(\mathfrak {F}_{hq}\)-supplemented in G if G has a quasinormal subgroup N such that HN is a Hall subgroup of G and \((H\cap N)H_{G}/ H_{G} \le Z_{\mathfrak {F}}(G/H_{G})\), where \(H_{G}\) is the core of H in G and \({Z}_{\mathfrak {F}} (G/H_{G})\) is the \(\mathfrak {F}\)-hypercenter of \({G/H}_{G}\). This paper concerns the structure of a finite group G under the assumption that some subgroups of G are \(\mathfrak {F}_{hq}\)-supplemented in G.     

Complementability conditions for a periodic almost solvable subgroup in the group containing it

It is proved that if every prime Sylow subgroup of a periodic almost solvable (more generally, periodic W₀-) subgroup H of a group G has a complement in G and if, moreover, H is at most countable and the set (H) is finite, the subgroup H itself possesses a complement in G.Deceased.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 6, pp. 822–826, June, 1992.     

K-invariants in the universal enveloping algebra of the desitter group

Let G be a non-compact connected semisimple Lie group with finite center and let G^K denote the centralizer of a maximal compact subgroup K of G inG, the universal enveloping algebra over of the Lie algebra of G. In [4] Lepowsky defines an injective anti-homo morphism P:G ^KK ^MA, where M is the centralizer in K of a Cartan subalgebraa of the symmetric pair (G,K),K andA are the universal enveloping algebras over corresponding to K anda, respectively, andK ^M is the centralizer of M inK. The subalgebra P(G ^K) ofK ^MA has considerable significance in the infinite dimensional representation theory of G. In this paper we explicitly compute P(G ^K) when G=S0_o(4,1), and show how this result leads to the determination of all irreducible representations of G and its universal covering group Spin(4,1).Partially supported by CONICET (Argentina) grants.     

Influence of properties of maximal subgroups on the structure of a finite group

We establish some tests for the solvability of finite groups and describe one class of unsolvable groups. We prove that an unsolvable group G such that a maximal subgroup M= P × H is nilpotent and the 2-Sylow subgroup P of M is metacyclic has a normal series GD-g₀TD-{1} such that T is contained in M, G₀/T - PSL (2, q), where q is a power of a prime of the form 2ⁿ± 1 and the index of g₀ in G is not greater than 2.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 183–190, February, 1972.The author warmly thanks V. A. Vedernikov for his guidance and assistance in completing this work.     

The semigroup of a strongly connected automaton

We show that each member of a class of strongly connected automata (containing all the finite ones) whose input semigroup is the semigroup of the automaton is isomorphic to an automaton (A,S,) where S is a semigroup of row-monomial matrices over a group G, A is a set of equivalence classes of monomial vectors over G and is the usual product of a matrix acting on a vector.     

Lattice of subgroups

Let D be a subgroup of the group G. The lattice of intermediate subgroups is studied. The subgroup F (D F G) is said to be D-complete, if D^F=u:u F>=F. Let F be the subset of all D-complete intermediate subgroups. The system {F, N_G(F)} is a fan for D in G (RZhMat, 1980, 5A208) if and only if D^x> is a D-complete subgroup for any x G. The set {F_ga} coincides with the collection of subgroups of the form D^A (1 A C G) if and only if for any x G the subgroup D, D^x is D-complete. The last condition holds, for example, for a pronormal subgroup D.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 13–19, 1980.     

On Semigroup Presentations and Efficiency

The purpose of this paper is twofold. First we determine some forms of the relations in a finite semigroup presentation with zero deficiency which does or does not define a group. Moreover, we conclude that a finite Rees matrix semigroup M [G; I, Λ; P] is efficient when G is efficient and the index sets I, Λ are finite.     

The inverse hull of the free semigroup on a set X

In this paper the inverse hull of the free semigroup on a non-empty set X is characterized. It is shown that if F^O is a certain semilattice and (G,*) is the free group on X, then is isomorphic to a Rees factor semigroup of a subsemigroup of (F^O x G, #), where (A,w)#(B,u) = ((A*w B)*w^–1, w*u). Several properties of , including Green's relations on are also determined.     

Characterizing cryptogroups with a finite number of $${\mathcal{H}}$$ -classes in each $${\mathcal{D}}$$ -class by their subsemigroups

We construct a family of completely regular semigroups with the property that each completely regular semigroup S with a finite number of -classes in each -class is non-cryptic if and only if S contains an isomorphic image of a member of . Each member F of is an ideal extension of a Rees matrix semigroup J by a cyclic group B with a zero adjoined and the identity of B is the identity of F. Here with I and Λ finite, G is given by generators and relations, and P is given explicitly. Within completely regular semigroups, the cryptic property is equivalent to where is the natural partial order and a if and only if a ² = ab = ba. Hence the above result can be formulated in terms of and .      

Groups with Largely Splitting Automorphisms of Orders Three and Four

A subset X of a group G is said to be large (on the left) if, for any finite set of elements g₁,l... ,g_kin G, an intersection of the subsets g_iX=g_imid x in X is not empty, that is, limits_{i=1} ^{k}g_iX . It is proved that a group in which elements of order 3 form a large subset is in fact of exponent 3. This result follows from the more general theorem on groups with a largely splitting automorphism of order 3, thus answering a question posed by Jaber amd Wagner in [1]. For groups with a largely splitting automorphism of order 4, it is shown that if His a normal -invariant soluble subgroup of derived length d then the derived subgroup [H,H] is nilpotent of class bounded in terms of d. The special case where =1 yields the same result for groups that are largely of exponent 4.     

A canonical form for self-adjoint pencils in Hilbert space

We characterize those pencils P=A–B of operators on a separable Hilbert space H for which a linear homeomorphism D of H exists satisfying the following: (i) H decomposes into a direct sum F+G, orthogonal in a (perhaps indefinite) inner product induced by A, where F is finite dimensional (ii) D^*PD=P_F(I_G–C) where P_F is a (congruence) canonical form for the general self-adjoint pencil on F, and C is a bounded self-adjoint operator on G. For a given P, explicit constructions are given for C, D, F, G and P_F.     

Polynomidentitäten und Darstellungen von Gruppen

Generalizing results in [1], [3], [5], and [7] it is shown that the algebra generated by the quasi-regular representation of a discrete groupG onl ² (G/H) satisfies a standard polynomial identity if and only if (G) contains an abelian normal subgroup of finite index. An analogon for topological groups is also proved.