Combinatorial Group Theory
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Frattini Subgroups and -Central Groups
Pacific Journal of Mathematics FRATTINI SUBGROUPS AND 8-CENTRAL GROUPS HOMER FRANKLIN BECHTELL,JR. Vol. 18, No. 1 March 1966 PACIFIC JOURNAL OF MATHEMATICS Vol. 18, No. 1, 1966 FRATTINI SUBGROUPS AND φ-CENTRAL GROUPS HOMER BECHTELL 0-central groups are introduced as a step In the direction of determining sufficiency conditions for a group to be the Frattini subgroup of some unite p-gronp and the related exten- sion problem. The notion of Φ-centrality arises by uniting the concept of an E-group with the generalized central series of Kaloujnine. An E-group is defined as a finite group G such that Φ(N) ^ Φ(G) for each subgroup N ^ G. If Sίf is a group of automorphisms of a group N, N has an i^-central series ι a N = No > Nt > > Nr = 1 if x~x e N3- for all x e Nj-lf all a a 6 £%f, x the image of x under the automorphism a e 3ίf y i = 0,l, •••, r-1. Denote the automorphism group induced OR Φ(G) by trans- formation of elements of an £rgroup G by 3ίf. Then Φ{£ίf) ~ JP'iΦiG)), J^iβiG)) the inner automorphism group of Φ(G). Furthermore if G is nilpotent9 then each subgroup N ^ Φ(G), N invariant under 3ίf \ possess an J^-central series. A class of niipotent groups N is defined as ^-central provided that N possesses at least one niipotent group of automorphisms ££'' Φ 1 such that Φ{βίf} — ,J^(N) and N possesses an J^-central series. Several theorems develop results about (^-central groups and the associated ^^-central series analogous to those between niipotent groups and their associated central series. -
Filling Functions Notes for an Advanced Course on the Geometry of the Word Problem for Finitely Generated Groups Centre De Recer
Filling Functions Notes for an advanced course on The Geometry of the Word Problem for Finitely Generated Groups Centre de Recerca Mathematica` Barcelona T.R.Riley July 2005 Revised February 2006 Contents Notation vi 1Introduction 1 2Fillingfunctions 5 2.1 Van Kampen diagrams . 5 2.2 Filling functions via van Kampen diagrams . .... 6 2.3 Example: combable groups . 10 2.4 Filling functions interpreted algebraically . ......... 15 2.5 Filling functions interpreted computationally . ......... 16 2.6 Filling functions for Riemannian manifolds . ...... 21 2.7 Quasi-isometry invariance . .22 3Relationshipsbetweenfillingfunctions 25 3.1 The Double Exponential Theorem . 26 3.2 Filling length and duality of spanning trees in planar graphs . 31 3.3 Extrinsic diameter versus intrinsic diameter . ........ 35 3.4 Free filling length . 35 4Example:nilpotentgroups 39 4.1 The Dehn and filling length functions . .. 39 4.2 Open questions . 42 5Asymptoticcones 45 5.1 The definition . 45 5.2 Hyperbolic groups . 47 5.3 Groups with simply connected asymptotic cones . ...... 53 5.4 Higher dimensions . 57 Bibliography 68 v Notation f, g :[0, ∞) → [0, ∞)satisfy f ≼ g when there exists C > 0 such that f (n) ≤ Cg(Cn+ C) + Cn+ C for all n,satisfy f ≽ g ≼, ≽, ≃ when g ≼ f ,andsatisfy f ≃ g when f ≼ g and g ≼ f .These relations are extended to functions f : N → N by considering such f to be constant on the intervals [n, n + 1). ab, a−b,[a, b] b−1ab, b−1a−1b, a−1b−1ab Cay1(G, X) the Cayley graph of G with respect to a generating set X Cay2(P) the Cayley 2-complex of a -
18.703 Modern Algebra, Presentations and Groups of Small
12. Presentations and Groups of small order Definition-Lemma 12.1. Let A be a set. A word in A is a string of 0 elements of A and their inverses. We say that the word w is obtained 0 from w by a reduction, if we can get from w to w by repeatedly applying the following rule, −1 −1 • replace aa (or a a) by the empty string. 0 Given any word w, the reduced word w associated to w is any 0 word obtained from w by reduction, such that w cannot be reduced any further. Given two words w1 and w2 of A, the concatenation of w1 and w2 is the word w = w1w2. The empty word is denoted e. The set of all reduced words is denoted FA. With product defined as the reduced concatenation, this set becomes a group, called the free group with generators A. It is interesting to look at examples. Suppose that A contains one element a. An element of FA = Fa is a reduced word, using only a and a−1 . The word w = aaaa−1 a−1 aaa is a string using a and a−1. Given any such word, we pass to the reduction w0 of w. This means cancelling as much as we can, and replacing strings of a’s by the corresponding power. Thus w = aaa −1 aaa = aaaa = a 4 = w0 ; where equality means up to reduction. Thus the free group on one generator is isomorphic to Z. The free group on two generators is much more complicated and it is not abelian. -
Some Remarks on Semigroup Presentations
SOME REMARKS ON SEMIGROUP PRESENTATIONS B. H. NEUMANN Dedicated to H. S. M. Coxeter 1. Let a semigroup A be given by generators ai, a2, . , ad and defining relations U\ — V\yu2 = v2, ... , ue = ve between these generators, the uu vt being words in the generators. We then have a presentation of A, and write A = sgp(ai, . , ad, «i = vu . , ue = ve). The same generators with the same relations can also be interpreted as the presentation of a group, for which we write A* = gpOi, . , ad\ «i = vu . , ue = ve). There is a unique homomorphism <£: A—* A* which maps each generator at G A on the same generator at G 4*. This is a monomorphism if, and only if, A is embeddable in a group; and, equally obviously, it is an epimorphism if, and only if, the image A<j> in A* is a group. This is the case in particular if A<j> is finite, as a finite subsemigroup of a group is itself a group. Thus we see that A(j> and A* are both finite (in which case they coincide) or both infinite (in which case they may or may not coincide). If A*, and thus also Acfr, is finite, A may be finite or infinite; but if A*, and thus also A<p, is infinite, then A must be infinite too. It follows in particular that A is infinite if e, the number of defining relations, is strictly less than d, the number of generators. 2. We now assume that d is chosen minimal, so that A cannot be generated by fewer than d elements. -
Knapsack Problems in Groups
MATHEMATICS OF COMPUTATION Volume 84, Number 292, March 2015, Pages 987–1016 S 0025-5718(2014)02880-9 Article electronically published on July 30, 2014 KNAPSACK PROBLEMS IN GROUPS ALEXEI MYASNIKOV, ANDREY NIKOLAEV, AND ALEXANDER USHAKOV Abstract. We generalize the classical knapsack and subset sum problems to arbitrary groups and study the computational complexity of these new problems. We show that these problems, as well as the bounded submonoid membership problem, are P-time decidable in hyperbolic groups and give var- ious examples of finitely presented groups where the subset sum problem is NP-complete. 1. Introduction 1.1. Motivation. This is the first in a series of papers on non-commutative discrete (combinatorial) optimization. In this series we propose to study complexity of the classical discrete optimization (DO) problems in their most general form — in non- commutative groups. For example, DO problems concerning integers (subset sum, knapsack problem, etc.) make perfect sense when the group of additive integers is replaced by an arbitrary (non-commutative) group G. The classical lattice problems are about subgroups (integer lattices) of the additive groups Zn or Qn, their non- commutative versions deal with arbitrary finitely generated subgroups of a group G. The travelling salesman problem or the Steiner tree problem make sense for arbitrary finite subsets of vertices in a given Cayley graph of a non-commutative infinite group (with the natural graph metric). The Post correspondence problem carries over in a straightforward fashion from a free monoid to an arbitrary group. This list of examples can be easily extended, but the point here is that many classical DO problems have natural and interesting non-commutative versions. -
The Free Product of Groups with Amalgamated Subgroup Malnorrnal in a Single Factor
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF PURE AND APPLIED ALGEBRA Journal of Pure and Applied Algebra 127 (1998) 119-136 The free product of groups with amalgamated subgroup malnorrnal in a single factor Steven A. Bleiler*, Amelia C. Jones Portland State University, Portland, OR 97207, USA University of California, Davis, Davis, CA 95616, USA Communicated by C.A. Weibel; received 10 May 1994; received in revised form 22 August 1995 Abstract We discuss groups that are free products with amalgamation where the amalgamating subgroup is of rank at least two and malnormal in at least one of the factor groups. In 1971, Karrass and Solitar showed that when the amalgamating subgroup is malnormal in both factors, the global group cannot be two-generator. When the amalgamating subgroup is malnormal in a single factor, the global group may indeed be two-generator. If so, we show that either the non-malnormal factor contains a torsion element or, if not, then there is a generating pair of one of four specific types. For each type, we establish a set of relations which must hold in the factor B and give restrictions on the rank and generators of each factor. @ 1998 Published by Elsevier Science B.V. All rights reserved. 0. Introduction Baumslag introduced the term malnormal in [l] to describe a subgroup that intersects each of its conjugates trivially. Here we discuss groups that are free products with amalgamation where the amalgamating subgroup is of rank at least two and malnormal in at least one of the factor groups. -
2-SYLOW THEORY for GROUPS of FINITE MORLEY RANK Contents 0
2-SYLOW THEORY FOR GROUPS OF FINITE MORLEY RANK SALMAN SIDDIQI Abstract. We explore 2-Sylow theory in groups of finite Morley rank, de- veloping the language necessary to begin to understand the structure of such groups along the way, and culminating in a proof of a theorem of Borovik and Poizat that all Sylow 2-subgroups are conjugate in a group of finite Morley rank. Contents 0. Introduction 1 1. Morley Rank on Groups 2 2. Connectedness, Nilpotency and Definable Closures 6 3. 2-Sylow Theory 8 Acknowledgements 15 References 15 0. Introduction The analysis of groups of finite Morley rank begins with the following conjecture (often called the algebraicity conjecture) of Cherlin and Zil'ber. Conjecture 0.1 (Cherlin-Zil'ber). A simple !-stable group is algebraic over an algebraically closed field. Though we will not elaborate much further, it is known that all algebraic groups over algebraically closed fields are simple groups of finite Morley rank, with rank equal to their dimension. In turn, !-stable groups are also of finite Morley rank. If this conjecture were true, all three of these would be equivalent: simple !- stable groups, simple groups of finite Morley rank and simple algebraic groups over algebraically closed fields. This characterisation would single-handedly resolve almost all open problems in the field. There is also a weaker conjecture that simple groups of finite Morley rank are algebraic over algebraically closed fields, however, and this already poses signif- icant challenges. Borovik has provided a possible path to understanding groups of finite Morley rank, transferring tools and techniques from the classification of finite simple groups. -
Supplement on the Symmetric Group
SUPPLEMENT ON THE SYMMETRIC GROUP RUSS WOODROOFE I presented a couple of aspects of the theory of the symmetric group Sn differently than what is in Herstein. These notes will sketch this material. You will still want to read your notes and Herstein Chapter 2.10. 1. Conjugacy 1.1. The big idea. We recall from Linear Algebra that conjugacy in the matrix GLn(R) corresponds to changing basis in the underlying vector space n n R . Since GLn(R) is exactly the automorphism group of R (check the n definitions!), it’s equivalent to say that conjugation in Aut R corresponds n to change of basis in R . Similarly, Sn is Sym[n], the symmetries of the set [n] = {1, . , n}. We could think of an element of Sym[n] as being a “set automorphism” – this just says that sets have no interesting structure, unlike vector spaces with their abelian group structure. You might expect conjugation in Sn to correspond to some sort of change in basis of [n]. 1.2. Mathematical details. Lemma 1. Let g = (α1, . , αk) be a k-cycle in Sn, and h ∈ Sn be any element. Then h g = (α1 · h, α2 · h, . αk · h). h Proof. We show that g has the same action as (α1 · h, α2 · h, . αk · h), and since Sn acts faithfully (with trivial kernel) on [n], the lemma follows. h −1 First: (αi · h) · g = (αi · h) · h gh = αi · gh. If 1 ≤ i < m, then αi · gh = αi+1 · h as desired; otherwise αm · h = α1 · h also as desired. -
Combinatorial Group Theory
Combinatorial Group Theory Charles F. Miller III March 5, 2002 Abstract These notes were prepared for use by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996. They have subsequently been updated for use by students in the subject 620-421 Combinatorial Group Theory at the University of Melbourne. Copyright 1996-2002 by C. F. Miller. Contents 1 Free groups and presentations 3 1.1 Free groups . 3 1.2 Presentations by generators and relations . 7 1.3 Dehn’s fundamental problems . 9 1.4 Homomorphisms . 10 1.5 Presentations and fundamental groups . 12 1.6 Tietze transformations . 14 1.7 Extraction principles . 15 2 Construction of new groups 17 2.1 Direct products . 17 2.2 Free products . 19 2.3 Free products with amalgamation . 21 2.4 HNN extensions . 24 3 Properties, embeddings and examples 27 3.1 Countable groups embed in 2-generator groups . 27 3.2 Non-finite presentability of subgroups . 29 3.3 Hopfian and residually finite groups . 31 4 Subgroup Theory 35 4.1 Subgroups of Free Groups . 35 4.1.1 The general case . 35 4.1.2 Finitely generated subgroups of free groups . 35 4.2 Subgroups of presented groups . 41 4.3 Subgroups of free products . 43 4.4 Groups acting on trees . 44 5 Decision Problems 45 5.1 The word and conjugacy problems . 45 5.2 Higman’s embedding theorem . 51 1 5.3 The isomorphism problem and recognizing properties . 52 2 Chapter 1 Free groups and presentations In introductory courses on abstract algebra one is likely to encounter the dihedral group D3 consisting of the rigid motions of an equilateral triangle onto itself. -
Classification and Statistics of Finite Index Subgroups in Free Products
CLASSIFICATION AND STATISTICS OF FINITE INDEX SUBGROUPS IN FREE PRODUCTS Thomas W. Muller¨ and Jan-Christoph Schlage-Puchta 1. introduction ∗e For positive integers e and m denote by Cm the free product of e copies of the cyclic group of order m, and let Fr be the free group of rank r. Given integers r, t ≥ 0, distinct primes p1, . , pt, and positive integers e1, . , et, let ∗e1 ∗et Γ = Cp1 ∗ · · · ∗ Cpt ∗ Fr. (1) By the Kurosh subgroup theorem, a finite index subgroup ∆ ≤ Γ is again of the same ∼ ∗λ1 ∗λt form, that is, ∆ = Cp1 ∗ · · · ∗ Cpt ∗ Fµ with non-negative integers λ1, . , λt, µ. An Euler characteristic computation shows that the latter parameters are related to the index (Γ : ∆) via the relation X 1 X 1 λ 1 − + µ − 1 = (Γ : ∆) 1 − + r − 1 . (2) j p p j j j j The tuple τ(∆) := (λ1, . , λt; µ) is called the (isomorphism) type of ∆. The principal theme of the present paper is the enumeration of finite index subgroups ∆ in Γ under restrictions on τ(∆). In particular, we shall discuss, for Γ as above, the following three basic problems. (I) (Realization) Which abstract groups admitted by the Kurosh subgroup theorem are realized as finite index subgroups of Γ? (II) (Asymptotics) Find natural deformation conditions on τ ∈ Rt+1 implying an interesting asymptotic behaviour of the function sτ (Γ) counting the number of finite index subgroups in Γ of type τ. (III) (Distribution) What can we say about the distribution of isomorphism types for subgroups of index n in Γ (with respect to various weight distributions) as n tends to infinity? The motivation for these questions comes from three main sources: number theory, geometric function theory, and the theory of subgroup growth. -
On the Lattice of Subgroups of a Free Group: Complements and Rank
journal of Groups, Complexity, Cryptology Volume 12, Issue 1, 2020, pp. 1:1–1:24 Submitted Sept. 11, 2019 https://gcc.episciences.org/ Published Feb. 29, 2020 ON THE LATTICE OF SUBGROUPS OF A FREE GROUP: COMPLEMENTS AND RANK JORDI DELGADO AND PEDRO V. SILVA Centro de Matemática, Universidade do Porto, Portugal e-mail address: [email protected] Centro de Matemática, Universidade do Porto, Portugal e-mail address: [email protected] Abstract. A ∨-complement of a subgroup H 6 Fn is a subgroup K 6 Fn such that H ∨ K = Fn. If we also ask K to have trivial intersection with H, then we say that K is a ⊕-complement of H. The minimum possible rank of a ∨-complement (resp., ⊕-complement) of H is called the ∨-corank (resp., ⊕-corank) of H. We use Stallings automata to study these notions and the relations between them. In particular, we characterize when complements exist, compute the ∨-corank, and provide language-theoretical descriptions of the sets of cyclic complements. Finally, we prove that the two notions of corank coincide on subgroups that admit cyclic complements of both kinds. 1. Introduction Subgroups of free groups are complicated. Of course not the structure of the subgroups themselves (which are always free, a classic result by Nielsen and Schreier) but the relations between them, or more precisely, the lattice they constitute. A first hint in this direction is the fact that (free) subgroups of any countable rank appear as subgroups of the free group of rank 2 (and hence of any of its noncyclic subgroups) giving rise to a self-similar structure. -
Quasi P Or Not Quasi P? That Is the Question
Rose-Hulman Undergraduate Mathematics Journal Volume 3 Issue 2 Article 2 Quasi p or not Quasi p? That is the Question Ben Harwood Northern Kentucky University, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Harwood, Ben (2002) "Quasi p or not Quasi p? That is the Question," Rose-Hulman Undergraduate Mathematics Journal: Vol. 3 : Iss. 2 , Article 2. Available at: https://scholar.rose-hulman.edu/rhumj/vol3/iss2/2 Quasi p- or not quasi p-? That is the Question.* By Ben Harwood Department of Mathematics and Computer Science Northern Kentucky University Highland Heights, KY 41099 e-mail: [email protected] Section Zero: Introduction The question might not be as profound as Shakespeare’s, but nevertheless, it is interesting. Because few people seem to be aware of quasi p-groups, we will begin with a bit of history and a definition; and then we will determine for each group of order less than 24 (and a few others) whether the group is a quasi p-group for some prime p or not. This paper is a prequel to [Hwd]. In [Hwd] we prove that (Z3 £Z3)oZ2 and Z5 o Z4 are quasi 2-groups. Those proofs now form a portion of Proposition (12.1) It should also be noted that [Hwd] may also be found in this journal. Section One: Why should we be interested in quasi p-groups? In a 1957 paper titled Coverings of algebraic curves [Abh2], Abhyankar conjectured that the algebraic fundamental group of the affine line over an algebraically closed field k of prime characteristic p is the set of quasi p-groups, where by the algebraic fundamental group of the affine line he meant the family of all Galois groups Gal(L=k(X)) as L varies over all finite normal extensions of k(X) the function field of the affine line such that no point of the line is ramified in L, and where by a quasi p-group he meant a finite group that is generated by all of its p-Sylow subgroups.