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).)

/pdf/character-theory-of-finite-groups-mraeb8b123.pdf

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advances made, give a new approach to some classic topics, shed light on some fundamental facts that still remain unpublished and present some new subjects of research in the theory of classes of finite, not necessarily solvable, groups.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 survey article is to present some structural results about of groups whose Sylow p-subgroups are metacylic (p a prime)‎. ‎A complete characterisation of non-nilpotent groups whose 2-generator subgroups are metacyclic is also presented‎.

http://ijgt.ui.ac.ir/article_21480_00a08e7598db6145b7dad984861be684.pdf

On metacyclic subgroups of finite groups

A subgroup H of a finite group G is said to be τ-semipermutable in G if it permutes with all the Sylow subgroups Q of G such that (|H|, |Q|) = 1 and (|H|, |QG|) ≠ 1. A rather remarkable result of Lukyanenko and Skiba (Rend Semin Mat Univ Padova, 124:231–246, 2010) is: a finite solvable group G is a PST-group if and only if every subgroup of Fit(G) is τ-semipermutable. A local version of this result is established in this paper. A subgroup H of G is said to be τ-seminormal provided that it is normalized by all Sylow subgroups Q such that (|H|, |Q|) = 1 and (|H|, |QG|) ≠ 1. It is shown that a finite solvable group is a PST-group if and only if every subgroup of Fit(G) is τ-seminormal in G.

Some new characterizations of solvable PST-groups

https://www.degruyter.com/view/j/jgth.2000.3.issue-4/jgth.2000.030/jgth.2000.030.pdf

A note on m-permutable products of finite groups

In a recent paper, Berkovich studied how to describe the nilpotent residual of a group in terms of the nilpotent residuals of some of its subgroups. That study required the knowledge of the structure of the minimal nonnilpotent groups, also called Schmidt groups. The major aim of this paper is to show that this description could be obtained as a consequence of a more complete property, giving birth to some interesting generalizations. This purpose naturally led us to the study of a family of subgroup-closed saturated formations of nilpotent type. An innovative approach to these classes is provided.

On a theorem of Berkovich

All groups considered in this paper are finite. A subgroup~$H$ of a group~$G$ is said to \textit {seminormal} in $G$ if $H$ is normalized by all subgroups~$K$ of~$G$ such that $\gcd (\lvert H\rvert , \lvert K\rvert )=1$. We call a group $G$ an MSN-\textit {group} if the maximal subgroups of all the Sylow subgroups of~$G$ are seminormal in~$G$. In this paper, we classify all MSN-groups.

Adolfo Ballester-Bolinches

Papers

On metacyclic subgroups of finite groups

Some new characterizations of solvable PST-groups

A note on m-permutable products of finite groups

On a theorem of Berkovich

On seminormal subgroups of finite groups