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advances made, give a new approach to some classic topics, shed light on some fundamental facts that still remain unpublished and present some new subjects of research in the theory of classes of finite, not necessarily solvable, groups.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

In this survey paper several classes of finite soluble PST-groups are introduced and studied. We also determine how these classes are related.

Some special classes of finite soluble PST-groups

The class of all finite groups with nilpotent commutator subgroup is characterized as the largest subgroup-closed saturated formation 𝔉 for which the 𝔉-residual of a group generated by two 𝔉-subnormal subgroups is the subgroup generated by their 𝔉–residuals.

A Characterization of the Class of Finite Groups with Nilpotent Derived Subgroup

Let ℨ be a complete set of Sylow subgroups of a finite group G, that is, a set composed of a Sylow p-subgroup of G for each p dividing the order of G. A subgroup H of G is called ℨ-S-semipermutable if H permutes with every Sylow p-subgroup of G in ℨ for all p∉π(H); H is said to be ℨ-S-seminormal if it is normalized by every Sylow p-subgroup of G in ℨ for all p∉π(H). The main aim of this paper is to characterize the ℨ-MS-groups, or groups G in which the maximal subgroups of every Sylow subgroup in ℨ are ℨ-S-semipermutable in G and the ℨ-MSN-groups, or groups in which the maximal subgroups of every Sylow subgroup in ℨ are ℨ-S-seminormal in G.

On two classes of finite supersoluble groups

Assume that is a finite group and is a 2-nilpotent Sylow tower Hall subgroup of such that if and are -conjugate elements of of prime order or order 4, then and are -conjugate. We prove that there exists a normal subgroup of such that and .

A note on normal complements for finite groups

Under study are the factorizations of one-generated $ \mathfrak{X} $-local formations, where $ \mathfrak{X} $ is a class of simple groups. Some known results on one-generated local and Baer-local formations appear as particular cases.

Adolfo Ballester-Bolinches

Papers

Some special classes of finite soluble PST-groups

A Characterization of the Class of Finite Groups with Nilpotent Derived Subgroup

On two classes of finite supersoluble groups

A note on normal complements for finite groups

Factorizations of one-generated \( \mathfrak{X} \)-local formations