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advances made, give a new approach to some classic topics, shed light on some fundamental facts that still remain unpublished and present some new subjects of research in the theory of classes of finite, not necessarily solvable, groups.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

On the focal subgroup of a saturated fusion system

We analyze the influence of the maximal subgroups in the structure of a finite group G by means of the index complex introduced by Deskins in Proc. Sympos. Pure Math. vol. 1, Amer. Math. Soc. 1959, pp. 100-104.

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On the deskins index complex of a maximal subgroup of a finite group

On maximal subgroups of finite groups and theta pairs

In this paper, mutually sn-permutable subgroups of groups belonging to a class of generalised supersoluble groups are studied. Some analogs of known theorems on mutually sn-permutable products are established.

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On Products of Generalised Supersoluble Finite Groups

We give some sufficient conditions for a normal p-subgroup P of a finite group G to have every G-chief factor below it cyclic. The S-permutability of some p-subgroups of $${{\mathrm{O}}}^p(G)$$
 plays an important role. Some known results can be reproved and some others appear as corollaries of our main theorems.

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Adolfo Ballester-Bolinches

Papers

On the focal subgroup of a saturated fusion system

On the deskins index complex of a maximal subgroup of a finite group

On maximal subgroups of finite groups and theta pairs

On Products of Generalised Supersoluble Finite Groups

On the supersoluble hypercentre of a finite group