On bases of identities of finite central locally orthodox completely regular semigroups

Semigroup Forum - It has been known for a long time that every finite orthodox completely regular semigroup has a finite basis of identities, and that every finite central completely...     

On identities of finite involution semigroups

We investigate the question of whether a finite involution semigroup is inherently nonfinitely based (INFB), which means that it is not contained in any finitely based locally finite variety. Although we fall short of a full characterization, we nevertheless clarify a number of interesting subcases.     

On finite completely simple semigroups having no finite basis of identities

The purpose of this note is to provide a correct proof of the fact that there exist finite completely simple semigroups having no finite basis of identities.     

On finite hyperidentity bases for varieties of semigroups

For any varietyV of semigroups, we denote byH(V) the collection of all hyperidentities satisfied byV. It is natural to ask whether, for a givenV, H(V) is finitely based. This question has so far been answered, in the negative, for four varieties of semigroups: for the varieties of rectangular bands and of zero semigroups by the author in [8]; for the variety of semilattices by Penner in [5]; and for the varietyS of all semigroups by Bergman in [1]. In this paper, we show how Bergman's proof may in fact be used to deal with a large class of subvarieties ofS, namely all semigroup varieties except those satisfyingx ² =x ^2+m for somem. As a first step in the investigation of these exceptional varieties, we also present some hyperidentities satisfied by the variety B_1,1 of bands, and, using the same technique, show thatH(V) is not finitely based for any subvarietyV of B_1,1. These proofs all exploit the fact that the particular variety in question has hyperidentities of arbitrarily large arity. We conclude with an example of a variety for which even the collection of hyperidentities containing only one binary operation symbol is not finitely based.Presented by W. Taylor.Research supported by Natural Sciences & Engineering Research Council of Canada.     

Two-variable identities for finite solvable groups

We characterise the solvable groups in the class of finite groups by an inductively defined sequence of two-variable identities. Our main theorem is the analogue of a classical theorem of Zorn which gives a characterisation of the nilpotent groups in the class of finite groups by a sequence of two-variable identities. To cite this article: T. Bandman et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On intersections of solvable Hall subgroups in finite nonsolvable groups

The author continues the investigation of intersections of Hall subgroups in finite groups. Previously, the author proved that in the case when a Hall subgroup is Sylow there are three subgroups conjugate to it such that their intersection coincides with the maximal normal primary subgroup. A similar assertion holds for Hall subgroups in solvable groups. The aim of this paper is to construct examples of a (nonsolvable) group in which the intersection of any four subgroups conjugate to some Hall subgroup is nontrivial.     

On the existence of solvable normal subgroups in finite groups

The structure of a finite group in dependence on the structure of the subgroups generated by elements of its conjugate class is considered. Translated fromMatematischeskie Zametki, Vol. 61, No. 5, pp. 755–758, May, 1997. Translated by A. I. Shtern     

Finite subgroups of ∨-semilatticed inverse semigroups

In this paper we describe the ∨-semilatticed inverse semigroups generated by a finite group G. We do this by first constructing the free ∨-semilatticed inverse semigroup ?(G) on the group G. Each ∨-semilatticed inverse semigroup generated by G is a quotient of ?(G) by a ∨-congruence. We then describe these congruences and their corresponding quotients explicitly in terms of families of subgroups of G.     

Characters of finite inverse semigroups

For elements of a finite inverse semigroup, an equivalence relation called p-conugacy is introduced. It is proved that for any matrix representation of a finite inverse semigroup the values of the character of the representation are equal on p-conjugate elements. The number of inequivalent irreducible matrix representations of a finite inverse semigroup over the field of complex numbers is equal to the number of classes of p-conjugate elements.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 207–215, 1977.     

Factorizations of finite groups by solvable subgroups

This paper is concerned with finite groups whose composition factors are known simple groups.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 947–950, July–August, 1991.     

On the number of maximal subgroups of a finite solvable group

For a finite solvable group G and prime number p, we use elementary methods to obtain an upper bound for \mathfrak m_p(G){\mathfrak {m}_{p}(G)} , defined as the number of maximal subgroups of G whose index in G is a power of p. From this we derive an upper bound on the total number of maximal subgroups of a finite solvable group in terms of its order. This bound improves existing bounds, and we identify conditions on the order of a finite solvable group under which this bound is best possible.     

On finite solvable groups decomposable into the product of two nilpotent subgroups

We establish a series of results concerning various properties of a finite solvable groupG=AB with nilpotent subgroupsA andB. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 52, No. 6, pp. 809–819, June, 2000.     

On graph inverse semigroups

Semigroup Forum - We study the relationship between the graph inverse semigroups of two graphs when there is a directed immersion between the graphs and we provide structural information about...