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This page contains 4 mathematics books that I have writen. They can be purchased using Paypal. 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. I would urge anybody with editions I or II or III of the book to acquire edition IV - The Classification. 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, Russian editions of book III (electronic and paperback), as the area is popular in Russian speaking countries and, as noted, Russian language speakers have contributed positively in this area.(Currently these editions have been suspended due to collaboration delays). 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.
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