This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

1. (i) Suppose K is a conjugacy class of Sn contained in An; then K is called split if K is a union of two conjugacy classes of An. Show that the number of split conjugacy classes contained in An is equal to the number of characters χ ∈ Irr(Sn) such that χAn is not irreducible. (Hint. Consider the vector space of class functions on An which are invariant under conjugation by the transposition (12).)

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Character Theory of Finite Groups

The Algebraic Theory of Semigroups, Vol. I. By A. H. Clifford and G. B. Preston. Pp. xv + 224. $10.60. 1961. (American Mathematical Society, Providence.)

Many group theorists all over the world have been trying in the last twenty-five years to extend and adapt the magnificent methods of the Theory of Finite Soluble Groups to the more ambitious universe of all finite groups. This is a natural progression after the classification of finite simple groups but the achievements in this area are scattered in various papers.

Our objectives in this book were to gather, order and examine all this material, including the latest advances made, give a new approach to some classic topics, shed light on some fundamental facts that still remain unpublished and present some new subjects of research in the theory of classes of finite, not necessarily solvable, groups.

Classes of finite groups

C-Normality of Groups and Its Properties

The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called ${\mathcal{PT}}$-groups. In particular, it is shown that the finite solvable ${\mathcal{PT}}$-groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugate-permutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalently, in which every normal subgroup is permutable sensitive. However, there exist finite insolvable groups which are not ${\mathcal{PT}}$-groups but all subnormal subgroups of defect two are permutable.

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Permutable subnormal subgroups of finite groups

All groups considered in the paper will be finite and soluble.The class of groups whose members are a direct product of Hall subgroups cor-responding to a given partition of the set of all primes is a subgroup-closed saturatedformation enjoying many interesting properties of nilpotent type. In fact, this familyof formations was characterised in [2] as the subgroup-closed saturated formations Tfor which the set of all ^"-subnormal subgroups of each group G is a sublattice of thesubgroup lattice of G. Hence the members of this family are called lattice formations.The purpose of this paper is to prove that the majority of properties of nilpotenttype enjoyed by the lattice formations actually characterise them (Theorem 1).On the other hand, it is known that the Fitting subgroup of a group which is theproduct of two nilpotent subgroups inherits the factorisation [1, Lemma 2.5.7]. Thenatural question is now for which subgroup-closed Fitting formations T the ^-radical ofa group G which is the product of two ^-groups inherits the factorisation. As applicationof Theorem 1, we see that this property also characterises the aforesaid subgroup-closedFitting formations. This is Theorem 2 of the paper.We shall adhere to the notation and terminology employed in [5]. This book isour main reference for results concerning formations and Fitting classes. For the sakeof completeness, we gather some notation and recall some definitions and results. We

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/85BD27F9051DF0AAFA3E0D7100C2C12B/S0004972700018943a.pdf/div-class-title-nilpotent-like-fitting-formations-of-finite-soluble-groups-div.pdf

Nilpotent-like fitting formations of finite soluble groups

A subgroup $H$ of a finite group $G$ is said to be S-semipermutable in $G$ if $H$ permutes with every Sylow $q$-subgroup of $G$ for primes $q$ not dividing $|H$|. A finite group $G$ is an MS-group if the maximal subgroups of all the Sylow subgroups of $G$ are S-semipermutable in $G$. The aim of the present paper is to characterise the finite MS-groups.

DOI:

 10.1017/S0004972714000306

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On a class of supersoluble groups

If a subgroup H of a periodic group G satisfies HP = PH for all Sylow subgroups P of G, then we call H a Sylow-permutable, or S-permutable, subgroup of G. It is well known that S-permutability is not a transitive relation. In this paper, we study infinite periodic groups in which the relation to be S-permutable is transitive (PST-groups) and infinite periodic groups whose ascendant subgroups are S-permutable (ASP-groups).

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Infinite groups with Sylow permutable subgroups

Because of the isomorphism (X A) ! X = X ! (A ! X), the transition structure of a deterministic automaton with state set X and with inputs from an alphabet A can be viewed both as an algebra and as a coalgebra. This algebra-coalgebra duality goes back to Arbib and Manes, who formulated it as a duality between reachability and observability, and is ultimately based on Kalman’s duality in systems theory between controllability and observability. Recently, it was used to give a new proof of Brzozowski’s minimization algorithm for deterministic automata. Here we will use the algebra-coalgebra duality of automata as a common perspective for the study of both varieties and covarieties, which are classes of automata and languages dened by equations and coequations, respectively. We make a rst connection with Eilenberg’s denition of varieties of languages, which is based on the classical, algebraic notion of varieties of (transition) monoids.

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Adolfo Ballester-Bolinches

Papers

Permutable subnormal subgroups of finite groups

Nilpotent-like fitting formations of finite soluble groups

On a class of supersoluble groups

Infinite groups with Sylow permutable subgroups

Varieties and covarieties of languages (preliminary version)