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advances made, give a new approach to some classic topics, shed light on some fundamental facts that still remain unpublished and present some new subjects of research in the theory of classes of finite, not necessarily solvable, groups.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

Abstract If $$\sigma = \{ {\sigma }_{i} : i \in I \}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>i</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>I</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> is a partition of the set $$\mathbb {P}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> of all prime numbers, a subgroup H of a finite group G is said to be $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> - subnormal in G if H can be joined to G by means of a chain of subgroups $$H=H_{0} \subseteq H_{1} \subseteq \cdots \subseteq H_{n}=G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>⊆</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>⊆</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>⊆</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> such that either $$H_{i-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> normal in $$H_{i}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> or $$H_{i}/{{\,\mathrm{Core}\,}}_{H_{i}}(H_{i-1})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mrow> <mml:mspace /> <mml:mi>Core</mml:mi> <mml:mspace /> </mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is a $${\sigma }_{j}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:math> -group for some $$j \in I$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>I</mml:mi> </mml:mrow> </mml:math> , for every $$i=1, \ldots , n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> . If $$\sigma = \{\{2\}, \{3\}, \{5\}, ... \}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:mo>{</mml:mo> <mml:mn>2</mml:mn> <mml:mo>}</mml:mo> <mml:mo>,</mml:mo> <mml:mo>{</mml:mo> <mml:mn>3</mml:mn> <mml:mo>}</mml:mo> <mml:mo>,</mml:mo> <mml:mo>{</mml:mo> <mml:mn>5</mml:mn> <mml:mo>}</mml:mo> <mml:mo>,</mml:mo> <mml:mo>.</mml:mo> <mml:mo>.</mml:mo> <mml:mo>.</mml:mo> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> is the minimal partition, then the $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> -subnormality reduces to the classical subgroup embedding property of subnormality. A finite group X is said to be a Schmidt group if X is not nilpotent and every proper subgroup of X is nilpotent. Every non-nilpotent finite group G has Schmidt subgroups and a detailed knowledge of their embedding in G can provide a deep insight into its structure. In this paper, a complete description of a finite group with $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> -subnormal Schmidt subgroups is given. It answers a question posed by Guo, Safonova and Skiba. 

https://link.springer.com/content/pdf/10.1007/s40840-022-01369-y.pdf

Finite Groups with $$\sigma $$-Subnormal Schmidt Subgroups

In this paper, we investigate the structure of locally finite groups of infinite section rank (respectively, special rank) whose subgroups of infinite section rank (respectively, special rank) are Sylow permutable, permutable or normal. Some earlier results for locally finite groups appear as consequences of our study.

/pdf/on-groups-whose-subgroups-of-infinite-rank-are-sylow-866b67bbcs.pdf

On groups whose subgroups of infinite rank are Sylow permutable

Lower bounds for the number of elements of the largest non-commuting set of a finite soluble group with a CC subgroup are considered in this paper.

/pdf/on-non-commuting-sets-in-finite-soluble-cc-groups-1z5md9owc5.pdf

On non-commuting sets in finite soluble CC-groups

The main aim of this paper is to give structural information of a finite group of minimal order belonging to a subgroup-closed class of finite groups and whose p-length is greater than 1, p a prime number. Alternative proofs and improvements of recent results about the influence of minimal p-subgroups on the p-nilpotence and p-length of a finite group arise as consequences of our study.

/pdf/on-the-p-length-of-some-finite-p-soluble-groups-1b48y0vxfo.pdf

On the p-length of some finite p-soluble groups

We consider mutually permutable products G = AB of two nilpotent groups. The structure of the Sylow p-subgroups of its nilpotent residual is described.

Adolfo Ballester-Bolinches

Papers

Finite Groups with $$\sigma $$-Subnormal Schmidt Subgroups

On groups whose subgroups of infinite rank are Sylow permutable

On non-commuting sets in finite soluble CC-groups

On the p-length of some finite p-soluble groups

The structure of mutually permutable products of finite nilpotent groups