The Local Structure for Finite Groups with a Large P-subgroup PDF Download

Author: Ulrich Meierfrankenfeld
Publisher:
ISBN: 9781470429485
Category : Finite groups
Languages : en
Pages : 342

View

Book Description

Author: U. Meierfrankenfeld
Publisher: American Mathematical Soc.
ISBN: 1470418770
Category : Mathematics
Languages : en
Pages : 356

View

Book Description
Let p be a prime, G a finite Kp-group S a Sylow p-subgroup of G and Q a large subgroup of G in S (i.e., CG(Q)≤Q and NG(U)≤NG(Q) for 1≠U≤CG(Q)). Let L be any subgroup of G with S≤L, Op(L)≠1 and Q⋬L. In this paper the authors determine the action of L on the largest elementary abelian normal p-reduced p-subgroup YL of L.

Author:
Publisher: Elsevier
ISBN: 0080954138
Category : Mathematics
Languages : en
Pages : 223

View

Book Description
Locally Finite Groups

Author: Dipl.-Math. Felix F. Flemisch
Publisher: BoD – Books on Demand
ISBN: 3758321204
Category : Mathematics
Languages : en
Pages : 69

View

Book Description
This research paper continues [15]. We begin with giving a profound overview of the structure of arbitrary simple groups and in particular of the simple locally finite groups and reduce their Sylow theory for the prime p to a quite famous conjecture by Prof. Otto H. Kegel (see [37], Theorem 2.4: "Let the p-subgroup P be a p-uniqueness subgroup in the finite simple group S which belongs to one of the seven rank-unbounded families. Then the rank of S is bounded in terms of P.") about the rank-unbounded ones of the 19 known families of finite simple groups. We introduce a new scheme to describe the 19 families, the family T of types, define the rank of each type, and emphasise the rôle of Kegel covers. This part presents a unified picture of known results whose proofs are by reference. Subsequently we apply new ideas to prove the conjecture for the alternating groups. Thereupon we are remembering Kegel covers and *-sequences. Next we suggest a way 1) and a way 2) how to prove and even how to optimise Kegel's conjecture step-by-step or peu à peu which leads to Conjecture 1, Conjecture 2 and Conjecture 3 thereby unifying Sylow theory in locally finite simple groups with Sylow theory in locally finite and p-soluble groups whose joint study directs Sylow theory in (locally) finite groups. For any unexplained terminology we refer to [15]. We then continue the program begun above to optimise along the way 1) the theorem about the first type "An" of infinite families of finite simple groups step-by-step to further types by proving it for the second type "A = PSLn". We start with proving Conjecture 2 about the General Linear Groups over (commutative) locally finite fields, stating that their rank is bounded in terms of their p-uniqueness, and then break down this insight to the Special Linear Groups and the Projective Special Linear (PSL) Groups over locally finite fields. We close with suggestions for future research -> regarding the remaining rank-unbounded types (the "Classical Groups") and the way 2), -> regarding (locally) finite and p-soluble groups, and -> regarding Cauchy's and Galois' contributions to Sylow theory in finite groups. We much hope to enthuse group theorists with them. We include the predecessor research paper [15] as an Appendix.

Author: Dipl.-Math. Felix F. Flemisch
Publisher: BoD – Books on Demand
ISBN: 3754336428
Category : Mathematics
Languages : en
Pages : 26

View

Book Description
Part 2 of the Trilogy "Characterising Locally Finite Groups Satisfying the Strong Sylow Theorem for the Prime p" & "About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups" & "The Strong Sylow Theorem for the Prime p in Projective Special Linear Locally Finite Groups" is based on the author's research paper "About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups". This very beautiful and pioneering manuscript had been submitted for peer reviewing to the open access journals Advances in Group Theory and Applications (AGTA) (see https://www.advgrouptheory.com/ journal/) and Science Research Association (SCIREA) Journal of Mathematics (see https://www.scirea.org/ journal/Mathematics) but was very regrettably rejected by both of them (with ridiculous arguments). We first give a profound overview of the structure of simple groups and in particular of the simple locally finite groups and reduce their Sylow theory for the prime p to a famous conjecture of Prof. Otto H. Kegel (see [16], Theorem 2.4: "Let the p-subgroup P be a p-uniqueness subgroup in the finite simple group S which belongs to one of the seven rank-unbounded families. Then the rank of S is bounded in terms of P.") about the rank-unbounded ones of the 19 known families of finite simple groups. Part 2 introduces a new scheme to describe the 19 families, the family T of types, defines the rank of each type, and emphasises the rôle of Kegel covers. This part presents a unified picture of known results all proofs of which are by reference and it is the actual reason why our title starts with "About". We then apply beautiful new ideas to prove the conjecture for the alternating groups (see Page ii). Thereupon we are remembering Kegel covers and *-sequences. Finally we suggest a plan how to prove and even how to optimise the conjecture step-by-step or peu à peu which leads to further quite tough conjectures thereby unifying Sylow theory in locally finite simple groups with Sylow theory in locally finite and p-soluble groups. For any unexplained terminology we refer to [6].

Author: Dipl.-Math. Felix Flemisch
Publisher: BoD – Books on Demand
ISBN: 3754360876
Category : Mathematics
Languages : en
Pages : 118

View

Book Description
Part 1 of the Trilogy "Characterising Locally Finite Groups Satisfying the Strong Sylow Theorem for the Prime p" & "About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups" & "The Strong Sylow Theorem for the Prime p in Projective Special Linear Locally Finite Groups" is based on the beauteous BoD-Book "Characterising locally finite groups satisfying the strong Sylow Theorem for the prime p - Revised edition" (see ISBN 978-3-7562-3416-5) which in turn has been based on the author's research paper "Characterising Locally Finite Groups Satisfying the Strong Sylow Theorem for the Prime p" that was published on pp. 13-39 of Volume 13 of the gratifyingly open access mathematical journal Advances in Group Theory and Applications (AGTA) (see https://www.advgrouptheory.com/ journal/#read). Part 1 removes the highlights in light green of the Revised edition and adds the albeit considerably improved Pages i to vi, Pages 26a to 26f, and Pages xiii to xviii to the AGTA paper. In addition it adds the ten new Pages xv to xxiv to the Revised edition and thus renumbers the Pages xv to xviii into the Pages xxv to xxviii. It includes Reference [11] as Appendix 1 and Reference [10] as Appendix 2. Finally it calls to mind Prof. Otto H. Kegel's fine contribution to the conference Ischia Group Theory 2016.

Author: Dipl.-Math. Felix Flemisch
Publisher: BoD – Books on Demand
ISBN: 3756838927
Category : Mathematics
Languages : en
Pages : 26

View

Book Description
Part 2 of the Trilogy "Characterising Locally Finite Groups Satisfying the Strong Sylow Theorem for the Prime p" & "About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups" & "The Strong Sylow Theorem for the Prime p in Projective Special Linear Locally Finite Groups" is based on the author's research paper "About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups". This very beautiful and pioneering manuscript had been submitted for peer reviewing to the open access journals Advances in Group Theory and Applications (AGTA) (see https://www.advgrouptheory.com/journal/) and Science Research Association (SCIREA) Journal of Mathematics (see https://www.scirea.org/journal/Mathematics) but was very regrettably rejected by both of them (with ridiculous arguments). We first give a profound overview of the structure of simple groups and in particular of the simple locally finite groups and reduce their Sylow theory for the prime p to a famous conjecture of Prof. Otto H. Kegel (see [16], Theorem 2.4: "Let the p-subgroup P be a p-uniqueness subgroup in the finite simple group S which belongs to one of the seven rank-unbounded families. Then the rank of S is bounded in terms of P.") about the rank-unbounded ones of the 19 known families of finite simple groups. Part 2 introduces a new scheme to describe the 19 families, the family T of types, defines the rank of each type, and emphasises the rôle of Kegel covers. This part presents a unified picture of known results all proofs of which are by reference and it is the actual reason why our title starts with "About". We then apply beautiful new ideas to prove the conjecture for the alternating groups (see Page ii). Thereupon we are remembering Kegel covers and *-sequences. Finally we suggest a plan how to prove and even how to optimise the conjecture step-by-step or peu à peu which leads to further quite tough conjectures thereby unifying Sylow theory in locally finite simple groups with Sylow theory in locally finite and p-soluble groups. For any unexplained terminology we refer to [6].

Author: Dipl.-Math. Felix F. Flemisch
Publisher: BoD – Books on Demand
ISBN: 3756898539
Category : Mathematics
Languages : en
Pages : 48

View

Book Description
In Part 3 of the Trilogy "Characterising Locally Finite Groups Satisfying the Strong Sylow Theorem for the Prime p" & "About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups" & "The Strong Sylow Theorem for the Prime p in Projective Special Linear Locally Finite Groups" we continue the program begun in [10] to optimise along the way 1) its beautiful Theorem about the first type "An" of infinite families of finite simple groups step-by-step to further types by proving it for the second type "A = PSL n". We start with proving the Conjecture 2 of [10] about the General Linear Groups over (commutative) locally finite fields, stating that their rank is bounded in terms of their p-uniqueness, and then break down this insight to the Special Linear Groups and the Projective Special Linear (PSL) Groups over locally finite fields. We close with suggestions for future research regarding the remaining five rank-unbounded types (the "Classical Groups") and the way 2), regarding the (locally) finite and p-soluble groups, and regarding Augustin-Louis Cauchy's and Évariste Galois' contributions to Sylow theory in finite groups.

Author: Daniel Gorenstein
Publisher: American Mathematical Soc.
ISBN: 9780821803905
Category : Mathematics
Languages : en
Pages : 246

View

Book Description
The second volume of a series devoted to reorganizing and simplifying proof of the classification of the finite simple groups. In a single chapter, it lays the groundwork for the forthcoming analysis of finite simple groups, beginning with the theory of components, layers, and the generalized Fitting subgroup, which has been developed largely since Gorenstein's basic 1968 text and is now central to understanding the structure of finite groups. Suitable as an auxiliary text for a graduate course in group theory. Member prices are $35 for individual and $47 for institutions. Annotation copyright by Book News, Inc., Portland, OR

Author: Daniel Gorenstein
Publisher:
ISBN: 9781470406868
Category : Finite simple groups
Languages : en
Pages : 731

View

Book Description