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This page contains 4 mathematics books that I have writen. There is also a series of attractive papers related to the area of the formation generated by a finite group. They can be purchased in paperback booklet form using Paypal. The papers give original results in the area of formations of groups, specifically that of the formation generated by a finite group.
As well as being a useful repository of mathematical facts, the papers are also useful for learning methods and techniques in this area.
I would say that this area of mathematics is in its infancy in providing predictors of structural information about groups, in a different way to classical finite group theory.
I would also say that it is in its infancy in computer approaches to calculating subdirect subgroups and formations.
The series of articles can also be viewed as providing extension, clarification and amplication to some of the results in the books, particulalry book IV.

The buttons take you to Paypal's secure payment page. 

(1) "The Formation Generated By A Finite Group I"

(2) "The Formation Generated By A Finite Group II"

(3) "The Formation Generated By A Finite Group III - A Classification"

(4) "The Formation Generated By A Finite Group IV - The Classification"

These books contain results pertaining to the conjecture of the, post WWII German mathematician, Wolfgang Gaschütz: "Does the formation generated by a finite group contain only finitely many subformations?" These results extend many of the results in the author's PhD thesis.

However the conjecture itself is not true - a counter example was found in 2012, by the Russian/Belarussian mathematician, Vladimir Burichenko. The group giving rise to the counter example is of chief factor series length 3 with factors that alternate between abelian and non-abelian simple.

The books lead the reader from elementary notions, through a description with proofs, of all the necessary preliminaries, to a statement and proof of almost all the seminal cases in which the conjecture is true. Also included are descriptions of groups with infinitely many subformations and formation critical groups in their formation.
The finale of the series is book IV in which the theorem which classifies those groups whose formation contains only finitely many formation critical groups and sub-formations is given. The difference between the proofs given in book IV as opposed to book III, is that the proof in book IV does not rely in any way on the Bryant-Harschneck Theorem, a theorem which was key in early modern developments towards this classification.
The proof now is essentially distilled down to only using notions from 19th century mathematics (from Galois onwards), except for the defining notion of a formation itself which is of 20th century origin (Gaschütz, I believe, was key)

The books will be useful for someone wishing to work on the problem of the formation generated by a finite group, or questions in this area.

Of a new and emerging interest, is the use of formations to determine the existance, or not, of a group with a particular structure.

Editions I and II are only useful as a cost effective way for the (student) reader to get to grip with and see proofs of results in these related areas (classical group theory, modular representation theory and co-homology theory), as well as the beginning notions.

There are also currently a Russian edition of book III (electronic and paperback) in the making, as the area is popular in Russian speaking countries and, as noted, Russian language speakers have contributed positively in this area. (I don't know what's hapenning with the Russian collaborators but I have to say that this edition, which is based on the well proven Edition III, is not available. Seems to be some un-professionalism or bad leadership, that's all I can think).
Edition III includes 2 short one page inserts - (i) an errata of typing errors and (ii) an addendum, which establishes what is essentially the completion of 'The Classification': that is those finite groups whose formation contains only finitely many formation critical groups are also in one to one correspondence with those groups whose formation contains only finitely many subformations. These errata and additions are incorporated into edition IV.
Please contact me at 'paulfoy@mathematicalservices.co.uk' or +44 (0)7922113573 about these books. The Reader and the .pdf file are electronic downloads, valid for 7 days, from the date of purchase.

The Formation Generated By A Finite Group I
hardback book: (£5)
Location
Reader: (£5)
.pdf file: (£5)
The Formation Generated By A Finite Group II
hardback book (£10):
Location
Reader: (£10) .pdf file: (£10)
The Formation Generated By A Finite Group III
paperback book: (£20)
Reader: (£20) .pdf file: (£20)
The Formation Generated By A Finite Group III (Russian Edition)
paperback book: (£20)
Reader (Russian): (£20) .pdf file (Russian): (£20)
The Formation Generated By A Finite Group IV
paperback book:
(£30)

A few clarifications and errata for this book can be found here.
These should be read in combination with the papers on this page.
The Sub-Formations of a Group of Socle Length at most 2.
 
This paper (knowing already that they are finite in number) details explicitly the actual formation critical groups in the formation generated by such a group. From this it is easy to generate the subformations.

Paperback booklet, 24pp - £5.91 + P+P

Bon Appetit

This paper contains a series of miscellany, required to appreciate book IV thoroughly.
Of particular note is a proof of a corollory of the Bryant-Harschneck Theorem, completely independant of the theory of varieties. That is the Fitting subgroup belongs to the formation generated by the group to which it belongs.

Paperback booklet, 24pp - £5.92 + P+P

It's a good read.
The Sub-Formations of a Soluble group of Socle Length 3
 
This paper describes in detail the formation critical groups in a soluble group of socle length 3.
 
Paperback booklet, 16pp - £5.93 + P+P


Enjoy.
The Sub-Formations of a Soluble by Simple group of socle length 3.

This paper describes in detail the formation critical groups in a soluble by non abelian simple group of socle length 3.

Paperback booklet, 24pp - £5.94 + P+P

Tasty.
The Sub-Formations of a Soluble by Simple group.

This paper characterises the formation critical groups in a soluble by simple group. Both the cases of a soluble group and a soluble-by-non abelian simple group are considered.
This result is used in the Characterisation Theorem of book IV.

Paperback booklet, 20pp - £5.95 + P+P

A culmination.
A bundle of all 5 papers, specially discounted.

5 Paperback booklets, I-V, £30 + P+P

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