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Editorial board A Chief Editors: Drozd Yu.A. Kirichenko V.V. Sushchansky V.I. Institute of Mathematics Taras Shevchenko National Silesian University of NAS of Ukraine, Kyiv, University of Kyiv, Technology, UKRAINE UKRAINE POLAND [email protected] [email protected] [email protected] Vice Chief Editors: Komarnytskyj M.Ya. Petravchuk A.P. Zhuchok A.V. Lviv Ivan Franko Taras Shevchenko National Lugansk Taras Shevchenko University, UKRAINE University of Kyiv, National University, mykola_komarnytsky@ UKRAINE UKRAINE yahoo.com [email protected] [email protected] Scientific Secretaries: Babych V.M. Zhuchok Yu.V. Taras Shevchenko National Lugansk Taras Shevchenko University of Kyiv, UKRAINE National University, UKRAINE [email protected] [email protected] Editorial Board: Artamonov V.A. Marciniak Z. Sapir M. Moscow State Mikhail Warsaw University, Vanderbilt University, Lomonosov University, POLAND Nashville, TN, USA RUSSIA [email protected] [email protected] [email protected] Mazorchuk V. Shestakov I.P. Dlab V. University of Uppsala, University of Sao Paulo, Carleton University, SWEDEN BRAZIL Ottawa, CANADA [email protected] and Sobolev Institute of [email protected] Mathematics, Novosibirsk, RUSSIA Mikhalev A.V. [email protected] Futorny V.M. Moscow State Mikhail Sao Paulo University, Lomonosov University, BRAZIL RUSSIA Simson D. [email protected] [email protected] Nicholas Copernicus University, Torun, POLAND Grigorchuk R.I. Nekrashevych V. [email protected] Steklov Institute of Texas A&M University Mathematics, Moscow, College Station, RUSSIA TX, USA Subbotin I.Ya. [email protected], [email protected] College of Letters [email protected] and Sciences, National University, USA Olshanskii A.Yu. Kurdachenko L.A. [email protected] Vanderbilt University, Dnepropetrovsk University, Nashville, TN, USA UKRAINE Wisbauer R. alexander.olshanskiy@ [email protected] Heinrich Heine University, vanderbilt.edu Dusseldorf, GERMANY Kashu A.I. wisbauer@math. Pilz G. Institute of Mathematics uni-duesseldorf.de and Computer Science, Johannes Kepler AS of Moldova, Chisinau, University, Linz, Yanchevskii V.I. MOLDOVA AUSTRIA Institute of Mathematics [email protected] [email protected] NAS of Belarus, Minsk, BELARUS Lyubashenko V. Protasov I.V. [email protected] Institute of Mathematics Taras Shevchenko National NAS of Ukraine, Kyiv, University of Kyiv, Zelmanov E.I. UKRAINE UKRAINE University of California, [email protected] [email protected] San Diego, CA, USA [email protected] B Instructions for authors The aim of the journal “Algebra and Discrete Mathematics” (as ADM below) is to present timely the state-of-the-art accounts on modern research in all areas of algebra (general algebra, semigroups, groups, rings and modules, linear algebra, algebraic geometry, universal algebras, homological algebra etc.) and discrete mathematics (combinatorial analysis, graphs theory, mathematical logic, theory of automata, coding theory, cryptography etc.) Languages: Papers to be considered for publication in the ADM journal must be written in English. Preparing papers: A Papers submitted to the ADM journal should be prepared in LTEX. Authors are strongly encourages to prepare their papers using the ADM author package containing template, instructions, and ADM journal document class. ADM au- thor package is available from the journal web-site http://adm.luguniv.edu.ua. Graphical items should be prepared as eps (encapsulated PostScript) files and included by using the graphicx package. To avoid distortion from rescaling, figures must not be wider than 120 mm. Submitting papers: Authors who wish to submit their papers should use the submission system at our web-site http://admjournal.luguniv.edu.ua/. Please attach the source TeX-file of the paper as the main file and the PDF as a supplementary file during the submission. Submission of a manuscript implies that the work described has not been published before and that it is not under consideration for publication elsewhere. Required information: The following information is required with the submission (note that all contact information, particularly email addresses, must be supplied to avoid delay): 1) full postal address of each author; 2) e-mail of each author; 3) abstract (no more than 15 lines); 4) 2010 Mathematics subject classification (can be accessible from http://www.ams.org/msc); 5) key words and phrases. Proof-sheets: Authors receive only one set of proof-sheets in PDF format via e-mail for corrections. Only correction of misprints and minor changes can be made during proofreading. Alfred Lvovich Shmelkin (12.06.1938 – 22.12.2015) With deep sorrow and regret we learnt about the death on December 22, 2015 of our colleague, Alfred Lvovich Shmelkin, a prominent algebraist, Honoured Professor of Lomonosov Moscow State University and member of ADM editorial board over the past thirteen years. Alfred L. Shmelkin was born on June 12, 1938 in Moscow. In 1956 he began his studies at Moscow State University. He graduated in 1961 from the Department of Mathematics and Mechanics. For the next three years he was a PhD student of J.N. Golovin. In 1964 A.L. Shmelkin defended his PhD thesis in Algebra and three years after, in 1967, he got a degree of Doctor of Science for his dissertation entitled “Products of group varieties”. In 1972, by the age of 32, he already became a full Professor at the Department of Higher Algebra of the Lomonosov Moscow State University. D Alfred Lvovich Shmelkin (12.06.1938–22.12.2015) Professor Shmelkin devoted his life to science and teaching. As the founder and leader of the world famous school on group varieties, he made a great contribution to the theory of group varieties and infinite di- mensional Lie algebras. His main results, the famous Shmelkin-Neumann Theorem, the construction of verbal wreath products and group vari- eties of Lie type have found many applications and are widely used by many algebraists. Known for his outstanding teaching abilities, Professor Shmelkin shared his knowledge as a supervisor to 25 PhD students. Many of them became recognizable mathematicians: A. Olshanskii, Yu. Bakh- turin, Yu. Razmyslov, A. Krasilnikov, V. Spilrain, R. Stohr, holding the degree of Doctor of Science. We shall remember Professor Shmelkin as a talented mathematician, a mentor, and a kind, always ready to help, warm-hearted man. He will be greatly missed by all of us. The Editorial Board of Algebra and Discrete Mathematics Journal Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 21 (2016). Number 1, pp. 1–17 © Journal “Algebra and Discrete Mathematics” Classification of L-cross-sections of the finite symmetric semigroup up to isomorphism Eugenija Bondar∗ Communicated by V. Mazorchuk Abstract. Let Tn be the symmetric semigroup of full trans- formations on a finite set with n elements. In the paper we give a counting formula for the number of L-cross-sections of Tn and classify all L-cross-sections of Tn up to isomorphism. Introduction Let ρ be an equivalence relation on a semigroup S. A subsemigroup S′ of S is called a ρ-cross-section of S provided that S′ contains exactly one representative from each equivalence class of ρ. Thus, the restriction ρ to the subsemigroup S′ is the identity relation. It is natural to investigate the cross-sections with respect to equivalences related somehow to the semigroup operation: Green’s relations, conjugacy and various congruences. In general, a semigroup need not to have a ρ-cross-section. It is possible, for example, that a semigroup S has an R-cross-section, while L-cross- sections of S do not exist at all. Thus, the existence of cross-sections of a given semigroup is an essential and non-obvious fact. The transformation semigroups are classical objects for investigations in semigroup theory (see [1]). For the full finite symmetric semigroup Tn, all H- and R-cross-sections have been described in [3]. It has been proved ∗The author acknowledges support from the Ministry of Education and Science of the Russian Federation, project no. 1.1999.2014/K, and the Competitiveness Program of Ural Federal University. 2010 MSC: 20M20. Key words and phrases: symmetric semigroup, cross-section, Green’s relations. 2 Classification of L-cross-sections of Tn that there exists a unique R-cross-section up to isomorphism. A pair of non-isomorphic L-cross-sections of T4 has been constructed in [4]. The author has obtained a description of the L-cross-sections of Tn in [5] (see Theorem 1). In the present paper we continue to investigate L-cross-sections of Tn. We give necessary information in Section 1. Section 2 is devoted to some additional definitions. In Section 3 we show how to count all different L-cross-sections of Tn (Theorem 2). In Section 4 we classify all L-cross- sections up to isomorphism (Theorem 3). 1. Preliminaries For any nonempty set X, the set of all transformations of X into itself, written on the right, constitutes a semigroup under the composition x(αβ) = (xα)β for all x ∈ X. This semigroup is denoted by T(X) and called the symmetric semigroup. If |X| = n, then the symmetric semigroup T(X) is also denoted by Tn. We write idX for the identity transformation on X, and cx for the constant transformation whose image is the singleton {x}, x ∈ X. For the image of a transformation α ∈ Tn we write im (α). The cardinality | im (α)| of the image of α is called the rank of this transformation and is denoted by rk (α). The kernel of α is denoted by ker α. Recall that ker α = {(a, b) ∈ X × X | aα = bα}. If X′ ′ is a subset of X, then α|X′ is the restriction α to X . We will assume X is finite. As the nature of elements of X is not important for us, suppose further that X = {1, 2, .
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  • On a Class of Finite Soluble Groups

    On a Class of Finite Soluble Groups

    J. Group Theory 21 (2018), 839–846 DOI 10.1515/jgth-2018-0015 © de Gruyter 2018 On a class of finite soluble groups Adolfo Ballester-Bolinches, John Cossey and Yangming Li Communicated by Alexander Olshanskii Abstract. The aim of this paper is to study the class of finite groups in which every subgroup is self-normalising in its subnormal closure. It is proved that this class is a sub- group-closed formation of finite soluble groups which is not closed under taking Frattini extensions and whose members can be characterised by means of their Carter subgroups. This leads to new characterisations of finite soluble T-, PT- and PST-groups. Finite groups whose p-subgroups, p a prime, are self-normalising in their subnormal closure are also characterised. 1 Introduction All groups considered in this paper will be finite. It has long been known that permutable and subnormal subgroups play an important role in the structural study of the groups, and in recent years there has been a considerable interest in the phenomenon of subgroup permutability. Recall that a subgroup H is said to be permutable in a group G if HK KH for all D subgroups K of G, and Sylow permutable or S-permutable if HP PH for all D Sylow subgroups P of G. According to a known result of Kegel, S-permutable subgroups are always subnormal ([1, Theorem 1.2.14]), and the intersection of S-permutable (respectively, subnormal) subgroups of a group G is again S-per- mutable (respectively, subnormal) in G (see [1, Theorem 1.2.19] and [3, Corol- lary A.14.2]).
  • The F-Depth of an F-Projector

    The F-Depth of an F-Projector

    Pacific Journal of Mathematics THE Ᏺ-DEPTH OF AN Ᏺ-PROJECTOR H. J. SCHMIDT Vol. 46, No. 2 December 1973 PACIFIC JOURNAL OF MATHEMATICS Vol. 46, No. 2, 1973 THE g-DEPTH OF AN ^-PROJECTOR1 H. J. SCHMIDT, JR. Let % be a saturated formation and let G be a finite solvable group with ^-projector JP. In a fundamental work, Carter and Hawkes have shown that for suitably restricted % there is a chain of j^crucial maximal subgroups of G termi- nating with F. It is shown here that the number of links in such a chain is an ^-invariant of G, called the gf-depth of F in G and written d%(F, G). If 4(G) is the g-length of G then, provided § is normal subgroup-closed, the inequality S%(G) ^ 2 d%(F, G) + 1 is ob- tained. If F is also nilpotent of nilpotency class c(F), then it is proved that 4(G) ^ d%(F, G) + c(F). If % and ξ> are two such suitable saturated formations with § = &> comparisons of the invariants d%(F, G) and d$(H, G) are made, where F and H are respectively the $- and £>-pro- jectors of the the finite solvable group G. In particular, if H ^ F then dd(F, G) ^ d$(H, G), and if in addition d9(F, G) = d^H, G) then H = F. I* Introduction* In this paper all groups considered are finite and solvable. Throughout we let % be a saturated formation which is locally induced by a class of nonempty, integrated formations %(p), one for each prime p.