The Decomposability Problem for Torsion-Free Abelian Groups is Analytic-Complete

Size: px
Start display at page:

Download "The Decomposability Problem for Torsion-Free Abelian Groups is Analytic-Complete"

Transcription

1 The Decomposability Problem for Torsion-Free Abelian Groups is Analytic-Complete Kyle Riggs Indiana University April 29, 2014 Kyle Riggs Decomposability Problem 1/ 22

2 Computable Groups background A group (G, +) is computable if there is an enumeration ν : N G such that the set {(a, b, c) : ν(a) + ν(b) = ν(c)} is computable. Such an enumeration is called a computable presentation of G. This means statements of the form x + y = z and nx = y are computable for x, y, z G, n N. Kyle Riggs Decomposability Problem 2/ 22

3 Computable Groups background An abelian group G is decomposable if it has two nontrivial subgroups A and B such that G = A B. If not, it is indecomposable. Kyle Riggs Decomposability Problem 3/ 22

4 Computable Groups background An abelian group G is decomposable if it has two nontrivial subgroups A and B such that G = A B. If not, it is indecomposable. We would like to write a group as the direct sum of indecomposable subgroups. Can we classify the indecomposable groups? Kyle Riggs Decomposability Problem 3/ 22

5 Computable Groups background The only indecomposable torsion groups are the cocyclic groups (Z(p k ) for 1 k ). Kyle Riggs Decomposability Problem 4/ 22

6 Computable Groups background The only indecomposable torsion groups are the cocyclic groups (Z(p k ) for 1 k ). Every mixed group is decomposable. Kyle Riggs Decomposability Problem 4/ 22

7 Computable Groups background The only indecomposable torsion groups are the cocyclic groups (Z(p k ) for 1 k ). Every mixed group is decomposable. Torsion-free groups of rank 1 (subgroups of Q) are indecomposable. There is no other well-known class of indecomposable torsion-free abelian groups. This talk will explain why. Kyle Riggs Decomposability Problem 4/ 22

8 Computable Groups background The only indecomposable torsion groups are the cocyclic groups (Z(p k ) for 1 k ). Every mixed group is decomposable. Torsion-free groups of rank 1 (subgroups of Q) are indecomposable. There is no other well-known class of indecomposable torsion-free abelian groups. This talk will explain why. For computable torsion-free groups of finite rank (> 1), being decomposable is a Σ 0 3-complete property. Among computable groups of infinite rank, the property is Σ 1 1 -complete. Kyle Riggs Decomposability Problem 4/ 22

9 Here s an example of an indecomposable group of rank 2 (from Abelian Groups, L. Fuchs) : Kyle Riggs Decomposability Problem 5/ 22

10 Here s an example of an indecomposable group of rank 2 (from Abelian Groups, L. Fuchs) : Start with the free abelian group generated by two elements: x 1 and x 2. Add to the group the following elements: x 1 3, x 2 5, x 1 7, x 2 11, x 1 13, x Kyle Riggs Decomposability Problem 5/ 22

11 Here s an example of an indecomposable group of rank 2 (from Abelian Groups, L. Fuchs) : Start with the free abelian group generated by two elements: x 1 and x 2. Add to the group the following elements: x 1 3, x 2 5, x 1 7, x 2 11, x 1 13, x Finally, add the element x 1+x 2 2 to the group. Denote the group generated by these elements as G. Kyle Riggs Decomposability Problem 5/ 22

12 {x 1, x 2 } is still a basis for G because for every g G, there exist m, m 1, m 2 Z such that mg = m 1 x 1 + m 2 x 2 We can also say that there exist q 1, q 2 Q such that g = q 1 x 1 + q 2 x 2 However, this does not imply that the elements q 1 x 1 and q 2 x 2 exist. For example, x 1+x 2 2 = 1 2 x x 2. Kyle Riggs Decomposability Problem 6/ 22

13 Note that χ G (x 1 ) = (0, 1, 0, 1, 0, 1, 0,...) and χ G (x 2 ) = (0, 0, 1, 0, 1, 0, 1,...) But for any g = q 1 x 1 + q 2 x 2 with q 1, q 2 nonzero, t(g) = (0, 0, 0, 0, 0, 0,...) Kyle Riggs Decomposability Problem 7/ 22

14 If there are nontrivial subgroups A, B such that G = A B, then x 1 must be contained in a direct summand of this decomposition. This also holds for x 2. Kyle Riggs Decomposability Problem 8/ 22

15 If there are nontrivial subgroups A, B such that G = A B, then x 1 must be contained in a direct summand of this decomposition. This also holds for x 2. Because we added the element x 1+x 2 2 to the group, but neither x 1 2 nor x 2 2, then x 1 and x 2 must be in the same direct summand. Kyle Riggs Decomposability Problem 8/ 22

16 If there are nontrivial subgroups A, B such that G = A B, then x 1 must be contained in a direct summand of this decomposition. This also holds for x 2. Because we added the element x 1+x 2 2 to the group, but neither x 1 2 nor x 2 2, then x 1 and x 2 must be in the same direct summand. Thus, G = A (or B), so the group is indecomposable. We say the element x 1+x 2 2 serves as a link connecting x 1 and x 2. Kyle Riggs Decomposability Problem 8/ 22

17 In this example, x 1 and x 2 are elements of strictly maximal type. An element x has strictly maximal type if, for any y linearly independent from x, t(x) t(y) (there are infinitely many pairs p, k P N such that p k x but p k y). Kyle Riggs Decomposability Problem 9/ 22

18 In this example, x 1 and x 2 are elements of strictly maximal type. An element x has strictly maximal type if, for any y linearly independent from x, t(x) t(y) (there are infinitely many pairs p, k P N such that p k x but p k y). As we have seen, any element of strictly maximal type in a decomposable group must be contained in one of the direct summands. Kyle Riggs Decomposability Problem 9/ 22

19 In this example, x 1 and x 2 are elements of strictly maximal type. An element x has strictly maximal type if, for any y linearly independent from x, t(x) t(y) (there are infinitely many pairs p, k P N such that p k x but p k y). As we have seen, any element of strictly maximal type in a decomposable group must be contained in one of the direct summands. A group is indecomposable if it has a basis of elements of strictly maximal type with a link (or a chain of links) between every pair of basis elements. The converse of this statement does not hold. Kyle Riggs Decomposability Problem 9/ 22

20 If G is a torsion-free group of finite rank and G = A B, then for any bases ā, b of A, B (resp.) the set ā b is a basis for G with the following property: For every g G, if g = i q i a i + j r j b j, then there are elements a, b such that a = i q i a i and b = j r j b j Kyle Riggs Decomposability Problem 10/ 22

21 If G is a torsion-free group of finite rank and G = A B, then for any bases ā, b of A, B (resp.) the set ā b is a basis for G with the following property: For every g G, if g = i q i a i + j r j b j, then there are elements a, b such that a = i q i a i and b = j r j b j This allows us to talk about decomposability in terms of finite bases instead of subgroups. Kyle Riggs Decomposability Problem 10/ 22

22 For a computable group G there is a Π 0 2 sets such that formula BASIS on finite BASIS( x) x is a basis of G Kyle Riggs Decomposability Problem 11/ 22

23 For a computable group G there is a Π 0 2 sets such that formula BASIS on finite BASIS( x) x is a basis of G This gives us the following Σ 0 3 formula for decomposable torsion-free groups of finite rank: G is decomposable iff ( ā, b [G] <ω ) { BASIS(ā b) ā ø b ø ( y G) ( q Q <ω )( w G) [ ( q = ā + b y = q i a i + q j b j ) w = ] } q i a i i j i Kyle Riggs Decomposability Problem 11/ 22

24 To see this property is Σ 0 3-complete, we describe a computable function f : ω {torsion-free abelian groups of rank 2} such that G n is decomposable iff W n is cofinite. Kyle Riggs Decomposability Problem 12/ 22

25 Start with the group G in the previous example. G = x 1 + x 2, x 1 2 3, x 2 5, x 1 7, x 2 11,... Kyle Riggs Decomposability Problem 13/ 22

26 Start with the group G in the previous example. G = x 1 + x 2, x 1 2 3, x 2 5, x 1 7, x 2 11,... To create the group G n, we add to G the element such that Φ n (k). x 2 p 2k+1 for every k Kyle Riggs Decomposability Problem 13/ 22

27 Start with the group G in the previous example. G = x 1 + x 2, x 1 2 3, x 2 5, x 1 7, x 2 11,... To create the group G n, we add to G the element such that Φ n (k). x 2 p 2k+1 for every k If W n is coinfinite, there are still infinitely many primes dividing x 1 but not x 2, so the group remains indecomposable. Kyle Riggs Decomposability Problem 13/ 22

28 Start with the group G in the previous example. G = x 1 + x 2, x 1 2 3, x 2 5, x 1 7, x 2 11,... To create the group G n, we add to G the element such that Φ n (k). x 2 p 2k+1 for every k If W n is coinfinite, there are still infinitely many primes dividing x 1 but not x 2, so the group remains indecomposable. If W n is cofinite, then x 1 no longer has strictly maximal type, and a decomposition of G n exists. Kyle Riggs Decomposability Problem 13/ 22

29 Start with the group G in the previous example. G = x 1 + x 2, x 1 2 3, x 2 5, x 1 7, x 2 11,... To create the group G n, we add to G the element such that Φ n (k). x 2 p 2k+1 for every k If W n is coinfinite, there are still infinitely many primes dividing x 1 but not x 2, so the group remains indecomposable. If W n is cofinite, then x 1 no longer has strictly maximal type, and a decomposition of G n exists. Thus, the set of decomposable torsion-free groups of finite rank (> 1) is Σ 0 3 -complete. Kyle Riggs Decomposability Problem 13/ 22

30 If we adapt our previous formula to describe decomposable groups of infinite rank, we get the following: ( (ā b) [G] ω ) [BASIS(ā b) ā ø b ø ( y G) ( q Q <ω )( w G) (y = q i a i + q j b j w = q i a i )] i j i This is a Σ 1 1 formula, and in fact we will show that this property is Σ 1 1-complete for computable groups, and analytic-complete for groups in general. Kyle Riggs Decomposability Problem 14/ 22

31 We describe a (computable) function from (computable) trees in ω <ω to torsion-free abelian groups of infinite rank that gives us a group G T which is decomposable iff there is an infinite path through the tree T. Kyle Riggs Decomposability Problem 15/ 22

32 We start with a countable basis {x i } i 1, {x σ } σ ω <ω, {y j } j 1 and make each element of this basis infinitely divisible by a unique prime. Kyle Riggs Decomposability Problem 16/ 22

33 We start with a countable basis {x i } i 1, {x σ } σ ω <ω, {y j } j 1 and make each element of this basis infinitely divisible by a unique prime. Then, for each pair of x-elements we create a link using a unique prime. We do the same for each pair of y-elements. Kyle Riggs Decomposability Problem 16/ 22

34 We start with a countable basis {x i } i 1, {x σ } σ ω <ω, {y j } j 1 and make each element of this basis infinitely divisible by a unique prime. Then, for each pair of x-elements we create a link using a unique prime. We do the same for each pair of y-elements. For every n we also add an element of the form y 1 + y y n p f (n) Kyle Riggs Decomposability Problem 16/ 22

35 At this point, the group decomposes as A B, where A is the pure subgroup containing all the x-elements, and B the pure subgroup containing all the y-elements. Then, for every i we create a link between x i and y i. The group (denoted G) is now indecomposable. Kyle Riggs Decomposability Problem 17/ 22

36 Idea: Given a tree T, we add elements to G to create a group G T. If T has no infinite paths, G T is still indecomposable. If T has an infinite path π, G T = A T B π, where A T is the pure subgroup containing all the x-elements and B π is the pure subgroup containing all the elements of the form y n + x π n B π = y 1 + x σ1, y 2 + x σ2,... where σ n = n and σ n σ n+1 for all n. Kyle Riggs Decomposability Problem 18/ 22

37 Sketch: Let σ be a string of length n, and let p be the prime that infinitely divides y n. Whenever a string τ σ is found on T, we make x σ divisible by another power of p. Kyle Riggs Decomposability Problem 19/ 22

38 Sketch: Let σ be a string of length n, and let p be the prime that infinitely divides y n. Whenever a string τ σ is found on T, we make x σ divisible by another power of p. If there is a string of length n with infinitely many strings above it, then y n no longer has strictly maximal type. However, the set {y n } {x σ : σ = n ( τ T ) τ σ} does have strictly maximal type. This means that if y n decomposes as y n = a + b, then a and b must be in the span of this set. Kyle Riggs Decomposability Problem 19/ 22

39 When we find σ on T ( σ = n), then we break some links to allow y n + x σ to be contained in a direct summand. For example: x n + y n q = x n x σ q + y n + x σ q Kyle Riggs Decomposability Problem 20/ 22

40 When we find σ on T ( σ = n), then we break some links to allow y n + x σ to be contained in a direct summand. For example: x n + y n q = x n x σ q + y n + x σ q y 1 + y y n p f (n) = x σ 1 + x σ x σ p f (n) + (y 1 + x σ 1 ) + (y 2 + x σ 2 ) (y n + x σ ) p f (n) Kyle Riggs Decomposability Problem 20/ 22

41 We denote the finished group G T. If there is an infinite path π through T, G T = A T B π. If G T is decomposable, we can use the increasing sequence of links to find an infinite path through T. Kyle Riggs Decomposability Problem 21/ 22

42 We denote the finished group G T. If there is an infinite path π through T, G T = A T B π. If G T is decomposable, we can use the increasing sequence of links to find an infinite path through T. Thus, the set of computable decomposable torsion-free abelian groups of infinite rank is Σ 1 1-complete, and the set of decomposable torsion-free abelian groups of infinite rank is Σ 1 1 -complete. Kyle Riggs Decomposability Problem 21/ 22

43 Thank you! Kyle Riggs Decomposability Problem 22/ 22

Computable Properties of Decomposable. and Completely Decomposable Groups. Kyle Walter Riggs

Computable Properties of Decomposable. and Completely Decomposable Groups. Kyle Walter Riggs Computable Properties of Decomposable and Completely Decomposable Groups Kyle Walter Riggs Submitted to the faculty of the University Graduate School in partial fulfillment of the requirements for the

More information

The classification of torsion-free abelian groups up to isomorphism and quasi-isomorphism

The classification of torsion-free abelian groups up to isomorphism and quasi-isomorphism The classification of torsion-free abelian groups up to isomorphism and quasi-isomorphism Samuel Coskey Rutgers University Joint Meetings, 2008 Torsion-free abelian groups of finite rank A concise and

More information

The Number of Homomorphic Images of an Abelian Group

The Number of Homomorphic Images of an Abelian Group International Journal of Algebra, Vol. 5, 2011, no. 3, 107-115 The Number of Homomorphic Images of an Abelian Group Greg Oman Ohio University, 321 Morton Hall Athens, OH 45701, USA ggoman@gmail.com Abstract.

More information

II. Products of Groups

II. Products of Groups II. Products of Groups Hong-Jian Lai October 2002 1. Direct Products (1.1) The direct product (also refereed as complete direct sum) of a collection of groups G i, i I consists of the Cartesian product

More information

Section II.2. Finitely Generated Abelian Groups

Section II.2. Finitely Generated Abelian Groups II.2. Finitely Generated Abelian Groups 1 Section II.2. Finitely Generated Abelian Groups Note. In this section we prove the Fundamental Theorem of Finitely Generated Abelian Groups. Recall that every

More information

Section II.1. Free Abelian Groups

Section II.1. Free Abelian Groups II.1. Free Abelian Groups 1 Section II.1. Free Abelian Groups Note. This section and the next, are independent of the rest of this chapter. The primary use of the results of this chapter is in the proof

More information

Math 210B: Algebra, Homework 4

Math 210B: Algebra, Homework 4 Math 210B: Algebra, Homework 4 Ian Coley February 5, 2014 Problem 1. Let S be a multiplicative subset in a commutative ring R. Show that the localisation functor R-Mod S 1 R-Mod, M S 1 M, is exact. First,

More information

Generalized Pigeonhole Properties of Graphs and Oriented Graphs

Generalized Pigeonhole Properties of Graphs and Oriented Graphs Europ. J. Combinatorics (2002) 23, 257 274 doi:10.1006/eujc.2002.0574 Available online at http://www.idealibrary.com on Generalized Pigeonhole Properties of Graphs and Oriented Graphs ANTHONY BONATO, PETER

More information

Algebras with finite descriptions

Algebras with finite descriptions Algebras with finite descriptions André Nies The University of Auckland July 19, 2005 Part 1: FA-presentability A countable structure in a finite signature is finite-automaton presentable (or automatic)

More information

PROBLEMS FROM GROUP THEORY

PROBLEMS FROM GROUP THEORY PROBLEMS FROM GROUP THEORY Page 1 of 12 In the problems below, G, H, K, and N generally denote groups. We use p to stand for a positive prime integer. Aut( G ) denotes the group of automorphisms of G.

More information

June 7, 2007 THE ISOMORPHISM PROBLEM FOR TORSION-FREE ABELIAN GROUPS IS ANALYTIC COMPLETE.

June 7, 2007 THE ISOMORPHISM PROBLEM FOR TORSION-FREE ABELIAN GROUPS IS ANALYTIC COMPLETE. June 7, 2007 THE ISOMORPHISM PROBLEM FOR TORSION-FREE ABELIAN GROUPS IS ANALYTIC COMPLETE. ROD DOWNEY AND ANTONIO MONTALBÁN Abstract. We prove that the isomorphism problem for torsion-free Abelian groups

More information

1. Group Theory Permutations.

1. Group Theory Permutations. 1.1. Permutations. 1. Group Theory Problem 1.1. Let G be a subgroup of S n of index 2. Show that G = A n. Problem 1.2. Find two elements of S 7 that have the same order but are not conjugate. Let π S 7

More information

RIEMANN SURFACES. max(0, deg x f)x.

RIEMANN SURFACES. max(0, deg x f)x. RIEMANN SURFACES 10. Weeks 11 12: Riemann-Roch theorem and applications 10.1. Divisors. The notion of a divisor looks very simple. Let X be a compact Riemann surface. A divisor is an expression a x x x

More information

ON THE SUBGROUPS OF TORSION-FREE GROUPS WHICH ARE SUBRINGS IN EVERY RING

ON THE SUBGROUPS OF TORSION-FREE GROUPS WHICH ARE SUBRINGS IN EVERY RING italian journal of pure and applied mathematics n. 31 2013 (63 76) 63 ON THE SUBGROUPS OF TORSION-FREE GROUPS WHICH ARE SUBRINGS IN EVERY RING A.M. Aghdam Department Of Mathematics University of Tabriz

More information

ALGEBRA EXERCISES, PhD EXAMINATION LEVEL

ALGEBRA EXERCISES, PhD EXAMINATION LEVEL ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)

More information

Enumerations and Completely Decomposable Torsion-Free Abelian Groups

Enumerations and Completely Decomposable Torsion-Free Abelian Groups Enumerations and Completely Decomposable Torsion-Free Abelian Groups Alexander G. Melnikov Sobolev Institute of Mathematics, Novosibirsk, Russia vokinlem@bk.ru Abstract. We study possible degree spectra

More information

ADDITIVE GROUPS OF SELF-INJECTIVE RINGS

ADDITIVE GROUPS OF SELF-INJECTIVE RINGS SOOCHOW JOURNAL OF MATHEMATICS Volume 33, No. 4, pp. 641-645, October 2007 ADDITIVE GROUPS OF SELF-INJECTIVE RINGS BY SHALOM FEIGELSTOCK Abstract. The additive groups of left self-injective rings, and

More information

The MacWilliams Identities

The MacWilliams Identities The MacWilliams Identities Jay A. Wood Western Michigan University Colloquium March 1, 2012 The Coding Problem How to ensure the integrity of a message transmitted over a noisy channel? Cleverly add redundancy.

More information

Section Signed Measures: The Hahn and Jordan Decompositions

Section Signed Measures: The Hahn and Jordan Decompositions 17.2. Signed Measures 1 Section 17.2. Signed Measures: The Hahn and Jordan Decompositions Note. If measure space (X, M) admits measures µ 1 and µ 2, then for any α,β R where α 0,β 0, µ 3 = αµ 1 + βµ 2

More information

ABELIAN GROUPS WHOSE SUBGROUP LATTICE IS THE UNION OF TWO INTERVALS

ABELIAN GROUPS WHOSE SUBGROUP LATTICE IS THE UNION OF TWO INTERVALS J. Aust. Math. Soc. 78 (2005), 27 36 ABELIAN GROUPS WHOSE SUBGROUP LATTICE IS THE UNION OF TWO INTERVALS SIMION BREAZ and GRIGORE CĂLUGĂREANU (Received 15 February 2003; revised 21 July 2003) Communicated

More information

Linear Equations in Linear Algebra

Linear Equations in Linear Algebra Linear Equations in Linear Algebra.7 LINEAR INDEPENDENCE LINEAR INDEPENDENCE Definition: An indexed set of vectors {v,, v p } in n is said to be linearly independent if the vector equation x x x 2 2 p

More information

Problem 4 (Wed Jan 29) Let G be a finite abelian group. Prove that the following are equivalent

Problem 4 (Wed Jan 29) Let G be a finite abelian group. Prove that the following are equivalent Last revised: May 16, 2014 A.Miller M542 www.math.wisc.edu/ miller/ Problem 1 (Fri Jan 24) (a) Find an integer x such that x = 6 mod 10 and x = 15 mod 21 and 0 x 210. (b) Find the smallest positive integer

More information

Tight Subgroups in Almost Completely Decomposable Groups

Tight Subgroups in Almost Completely Decomposable Groups Journal of Algebra 225, 501 516 (2000) doi:10.1006/jabr.1999.8230, available online at http://www.idealibrary.com on Tight Subgroups in Almost Completely Decomposable Groups K. Benabdallah Département

More information

1 p mr. r, where p 1 < p 2 < < p r are primes, reduces then to the problem of finding, for i = 1,...,r, all possible partitions (p e 1

1 p mr. r, where p 1 < p 2 < < p r are primes, reduces then to the problem of finding, for i = 1,...,r, all possible partitions (p e 1 Theorem 2.9 (The Fundamental Theorem for finite abelian groups). Let G be a finite abelian group. G can be written as an internal direct sum of non-trival cyclic groups of prime power order. Furthermore

More information

NilBott Tower of Aspherical Manifolds and Torus Actions

NilBott Tower of Aspherical Manifolds and Torus Actions NilBott Tower of Aspherical Manifolds and Torus Actions Tokyo Metropolitan University November 29, 2011 (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29,

More information

Geometric Realization and K-Theoretic Decomposition of C*-Algebras

Geometric Realization and K-Theoretic Decomposition of C*-Algebras Wayne State University Mathematics Faculty Research Publications Mathematics 5-1-2001 Geometric Realization and K-Theoretic Decomposition of C*-Algebras Claude Schochet Wayne State University, clsmath@gmail.com

More information

Section III.15. Factor-Group Computations and Simple Groups

Section III.15. Factor-Group Computations and Simple Groups III.15 Factor-Group Computations 1 Section III.15. Factor-Group Computations and Simple Groups Note. In this section, we try to extract information about a group G by considering properties of the factor

More information

Graduate Preliminary Examination

Graduate Preliminary Examination Graduate Preliminary Examination Algebra II 18.2.2005: 3 hours Problem 1. Prove or give a counter-example to the following statement: If M/L and L/K are algebraic extensions of fields, then M/K is algebraic.

More information

Countable subgroups of Euclidean space

Countable subgroups of Euclidean space Countable subgroups of Euclidean space Arnold W. Miller April 2013 revised May 21, 2013 In his paper [1], Konstantinos Beros proved a number of results about compactly generated subgroups of Polish groups.

More information

Algebra. Travis Dirle. December 4, 2016

Algebra. Travis Dirle. December 4, 2016 Abstract Algebra 2 Algebra Travis Dirle December 4, 2016 2 Contents 1 Groups 1 1.1 Semigroups, Monoids and Groups................ 1 1.2 Homomorphisms and Subgroups................. 2 1.3 Cyclic Groups...........................

More information

Measurability Problems for Boolean Algebras

Measurability Problems for Boolean Algebras Measurability Problems for Boolean Algebras Stevo Todorcevic Berkeley, March 31, 2014 Outline 1. Problems about the existence of measure 2. Quests for algebraic characterizations 3. The weak law of distributivity

More information

DIVISIBLE MULTIPLICATIVE GROUPS OF FIELDS

DIVISIBLE MULTIPLICATIVE GROUPS OF FIELDS DIVISIBLE MULTIPLICATIVE GROUPS OF FIELDS GREG OMAN Abstract. Some time ago, Laszlo Fuchs asked the following question: which abelian groups can be realized as the multiplicative group of (nonzero elements

More information

Endotrivial modules. Nadia Mazza. June Lancaster University

Endotrivial modules. Nadia Mazza. June Lancaster University Endotrivial modules Nadia Mazza Lancaster University June 2010 Contents Endotrivial modules : background Analysis An exotic turn Final remarks Set-up p a prime; k = k a field of characteristic p; G a finite

More information

(9-5) Def Turing machine, Turing computable functions. (9-7) Def Primitive recursive functions, Primitive recursive implies Turing computable.

(9-5) Def Turing machine, Turing computable functions. (9-7) Def Primitive recursive functions, Primitive recursive implies Turing computable. (9-5) Def Turing machine, Turing computable functions. A.Miller M773 Computability Theory Fall 2001 (9-7) Def Primitive recursive functions, Primitive recursive implies Turing computable. 1. Prove f(n)

More information

Math 602, Fall 2002, HW 2, due 10/14/2002

Math 602, Fall 2002, HW 2, due 10/14/2002 Math 602, Fall 2002, HW 2, due 10/14/2002 Part A AI) A Fermat prime, p, is a prime number of the form 2 α + 1. E.g., 2, 3, 5, 17, 257,.... (a) Show if 2 α + 1 is prime then α = 2 β. (b) Say p is a Fermat

More information

Kac-Moody Algebras. Ana Ros Camacho June 28, 2010

Kac-Moody Algebras. Ana Ros Camacho June 28, 2010 Kac-Moody Algebras Ana Ros Camacho June 28, 2010 Abstract Talk for the seminar on Cohomology of Lie algebras, under the supervision of J-Prof. Christoph Wockel Contents 1 Motivation 1 2 Prerequisites 1

More information

On Compact Just-Non-Lie Groups

On Compact Just-Non-Lie Groups On Compact Just-Non-Lie Groups FRANCESCO RUSSO Mathematics Department of Naples, Naples, Italy E-mail:francesco.russo@dma.unina.it Abstract. A compact group is called a compact Just-Non-Lie group or a

More information

32 Divisibility Theory in Integral Domains

32 Divisibility Theory in Integral Domains 3 Divisibility Theory in Integral Domains As we have already mentioned, the ring of integers is the prototype of integral domains. There is a divisibility relation on * : an integer b is said to be divisible

More information

Construction of digital nets from BCH-codes

Construction of digital nets from BCH-codes Construction of digital nets from BCH-codes Yves Edel Jürgen Bierbrauer Abstract We establish a link between the theory of error-correcting codes and the theory of (t, m, s)-nets. This leads to the fundamental

More information

Fundamental Theorem of Finite Abelian Groups

Fundamental Theorem of Finite Abelian Groups Monica Agana Boise State University September 1, 2015 Theorem (Fundamental Theorem of Finite Abelian Groups) Every finite Abelian group is a direct product of cyclic groups of prime-power order. The number

More information

ROOT SYSTEMS AND DYNKIN DIAGRAMS

ROOT SYSTEMS AND DYNKIN DIAGRAMS ROOT SYSTEMS AND DYNKIN DIAGRAMS DAVID MEHRLE In 1969, Murray Gell-Mann won the nobel prize in physics for his contributions and discoveries concerning the classification of elementary particles and their

More information

COMMUTING ELEMENTS IN GALOIS GROUPS OF FUNCTION FIELDS. Fedor Bogomolov and Yuri Tschinkel

COMMUTING ELEMENTS IN GALOIS GROUPS OF FUNCTION FIELDS. Fedor Bogomolov and Yuri Tschinkel COMMUTING ELEMENTS IN GALOIS GROUPS OF FUNCTION FIELDS by Fedor Bogomolov and Yuri Tschinkel Abstract. We study the structure of abelian subgroups of Galois groups of function fields. Contents Introduction................................................

More information

THE CLASSIFICATION OF INFINITE ABELIAN GROUPS WITH PARTIAL DECOMPOSITION BASES IN L ω

THE CLASSIFICATION OF INFINITE ABELIAN GROUPS WITH PARTIAL DECOMPOSITION BASES IN L ω ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 47, Number 2, 2017 THE CLASSIFICATION OF INFINITE ABELIAN GROUPS WITH PARTIAL DECOMPOSITION BASES IN L ω CAROL JACOBY AND PETER LOTH ABSTRACT. We consider the

More information

Michael Giudici. on joint work with variously John Bamberg, Martin Liebeck, Cheryl Praeger, Jan Saxl and Pham Huu Tiep

Michael Giudici. on joint work with variously John Bamberg, Martin Liebeck, Cheryl Praeger, Jan Saxl and Pham Huu Tiep 3 2 -transitive permutation groups Michael Giudici Centre for the Mathematics of Symmetry and Computation on joint work with variously John Bamberg, Martin Liebeck, Cheryl Praeger, Jan Saxl and Pham Huu

More information

SOME USES OF SET THEORY IN ALGEBRA. Stanford Logic Seminar February 10, 2009

SOME USES OF SET THEORY IN ALGEBRA. Stanford Logic Seminar February 10, 2009 SOME USES OF SET THEORY IN ALGEBRA Stanford Logic Seminar February 10, 2009 Plan I. The Whitehead Problem early history II. Compactness and Incompactness III. Deconstruction P. Eklof and A. Mekler, Almost

More information

Part II. Logic and Set Theory. Year

Part II. Logic and Set Theory. Year Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 60 Paper 4, Section II 16G State and prove the ǫ-recursion Theorem. [You may assume the Principle of ǫ- Induction.]

More information

Bulletin of the Iranian Mathematical Society

Bulletin of the Iranian Mathematical Society ISSN: 1017-060X (Print) ISSN: 1735-8515 (Online) Special Issue of the Bulletin of the Iranian Mathematical Society in Honor of Professor Heydar Radjavi s 80th Birthday Vol 41 (2015), No 7, pp 155 173 Title:

More information

Group Algebras with the Bounded Splitting Property. Derek J.S. Robinson. Ischia Group Theory 2014

Group Algebras with the Bounded Splitting Property. Derek J.S. Robinson. Ischia Group Theory 2014 Group Algebras with the Bounded Splitting Property Derek J.S. Robinson University of Illinois at Urbana-Champaign Ischia Group Theory 2014 (Joint work with Blas Torrecillas, University of Almería) Derek

More information

2.3 The Krull-Schmidt Theorem

2.3 The Krull-Schmidt Theorem 2.3. THE KRULL-SCHMIDT THEOREM 41 2.3 The Krull-Schmidt Theorem Every finitely generated abelian group is a direct sum of finitely many indecomposable abelian groups Z and Z p n. We will study a large

More information

25.1 Ergodicity and Metric Transitivity

25.1 Ergodicity and Metric Transitivity Chapter 25 Ergodicity This lecture explains what it means for a process to be ergodic or metrically transitive, gives a few characterizes of these properties (especially for AMS processes), and deduces

More information

arxiv: v1 [math.gn] 28 Jun 2010

arxiv: v1 [math.gn] 28 Jun 2010 Locally minimal topological groups 2 To the memory of Ivan Prodanov (1935-1985) Lydia Außenhofer Welfengarten 1, 30167 Hannover, Germany. e-mail: aussenho@math.uni-hannover.de M. J. Chasco Dept. de Física

More information

Modules Over Principal Ideal Domains

Modules Over Principal Ideal Domains Modules Over Principal Ideal Domains Brian Whetter April 24, 2014 This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. To view a copy of this

More information

Injective Modules and Matlis Duality

Injective Modules and Matlis Duality Appendix A Injective Modules and Matlis Duality Notes on 24 Hours of Local Cohomology William D. Taylor We take R to be a commutative ring, and will discuss the theory of injective R-modules. The following

More information

ALGEBRA QUALIFYING EXAM PROBLEMS

ALGEBRA QUALIFYING EXAM PROBLEMS ALGEBRA QUALIFYING EXAM PROBLEMS Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND MODULES General

More information

EXTERIOR AND SYMMETRIC POWERS OF MODULES FOR CYCLIC 2-GROUPS

EXTERIOR AND SYMMETRIC POWERS OF MODULES FOR CYCLIC 2-GROUPS EXTERIOR AND SYMMETRIC POWERS OF MODULES FOR CYCLIC 2-GROUPS FRANK IMSTEDT AND PETER SYMONDS Abstract. We prove a recursive formula for the exterior and symmetric powers of modules for a cyclic 2-group.

More information

Generalized coclass trees

Generalized coclass trees University of Kentucky Zassenhaus 2007-03-17 Overview I am interested in a merger of the methods of finite p-groups and finite simple groups to understand perfect groups In this talk I describe some of

More information

PETER A. CHOLAK, PETER GERDES, AND KAREN LANGE

PETER A. CHOLAK, PETER GERDES, AND KAREN LANGE D-MAXIMAL SETS PETER A. CHOLAK, PETER GERDES, AND KAREN LANGE Abstract. Soare [23] proved that the maximal sets form an orbit in E. We consider here D-maximal sets, generalizations of maximal sets introduced

More information

4.2 Chain Conditions

4.2 Chain Conditions 4.2 Chain Conditions Imposing chain conditions on the or on the poset of submodules of a module, poset of ideals of a ring, makes a module or ring more tractable and facilitates the proofs of deep theorems.

More information

Selected exercises from Abstract Algebra by Dummit and Foote (3rd edition).

Selected exercises from Abstract Algebra by Dummit and Foote (3rd edition). Selected exercises from Abstract Algebra by Dummit and Foote (3rd edition). Bryan Félix Abril 12, 2017 Section 2.1 Exercise (6). Let G be an abelian group. Prove that T = {g G g < } is a subgroup of G.

More information

ZORN S LEMMA AND SOME APPLICATIONS

ZORN S LEMMA AND SOME APPLICATIONS ZORN S LEMMA AND SOME APPLICATIONS KEITH CONRAD 1. Introduction Zorn s lemma is a result in set theory that appears in proofs of some non-constructive existence theorems throughout mathematics. We will

More information

120A LECTURE OUTLINES

120A LECTURE OUTLINES 120A LECTURE OUTLINES RUI WANG CONTENTS 1. Lecture 1. Introduction 1 2 1.1. An algebraic object to study 2 1.2. Group 2 1.3. Isomorphic binary operations 2 2. Lecture 2. Introduction 2 3 2.1. The multiplication

More information

FIXED POINT SETS AND LEFSCHETZ MODULES. John Maginnis and Silvia Onofrei Department of Mathematics, Kansas State University

FIXED POINT SETS AND LEFSCHETZ MODULES. John Maginnis and Silvia Onofrei Department of Mathematics, Kansas State University AMS Sectional Meeting Middle Tennessee State University, Murfreesboro, TN 3-4 November 2007 FIXED POINT SETS AND LEFSCHETZ MODULES John Maginnis and Silvia Onofrei Department of Mathematics, Kansas State

More information

Math 121 Homework 5: Notes on Selected Problems

Math 121 Homework 5: Notes on Selected Problems Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements

More information

Introduction to modules

Introduction to modules Chapter 3 Introduction to modules 3.1 Modules, submodules and homomorphisms The problem of classifying all rings is much too general to ever hope for an answer. But one of the most important tools available

More information

arxiv: v2 [math.nt] 23 Sep 2011

arxiv: v2 [math.nt] 23 Sep 2011 ELLIPTIC DIVISIBILITY SEQUENCES, SQUARES AND CUBES arxiv:1101.3839v2 [math.nt] 23 Sep 2011 Abstract. Elliptic divisibility sequences (EDSs) are generalizations of a class of integer divisibility sequences

More information

TEST GROUPS FOR WHITEHEAD GROUPS

TEST GROUPS FOR WHITEHEAD GROUPS TEST GROUPS FOR WHITEHEAD GROUPS PAUL C. EKLOF, LÁSZLÓ FUCHS, AND SAHARON SHELAH Abstract. We consider the question of when the dual of a Whitehead group is a test group for Whitehead groups. This turns

More information

Maximal perpendicularity in certain Abelian groups

Maximal perpendicularity in certain Abelian groups Acta Univ. Sapientiae, Mathematica, 9, 1 (2017) 235 247 DOI: 10.1515/ausm-2017-0016 Maximal perpendicularity in certain Abelian groups Mika Mattila Department of Mathematics, Tampere University of Technology,

More information

JUMP DEGREES OF TORSION-FREE ABELIAN GROUPS

JUMP DEGREES OF TORSION-FREE ABELIAN GROUPS JUMP DEGREES OF TORSION-FREE ABELIAN GROUPS BROOKE M. ANDERSEN, ASHER M. KACH, ALEXANDER G. MELNIKOV, AND REED SOLOMON Abstract. We show, for each computable ordinal α and degree a > (α), the existence

More information

In memoriam of Michael Butler ( )

In memoriam of Michael Butler ( ) In memoriam of Michael Butler (1928 2012) Helmut Lenzing Universität Paderborn Auslander Conference 2013, Woods Hole, 19. April H. Lenzing (Paderborn) Michael Butler 1 / 1 ICRA Beijing (2000). M. Butler

More information

n [ F (b j ) F (a j ) ], n j=1(a j, b j ] E (4.1)

n [ F (b j ) F (a j ) ], n j=1(a j, b j ] E (4.1) 1.4. CONSTRUCTION OF LEBESGUE-STIELTJES MEASURES In this section we shall put to use the Carathéodory-Hahn theory, in order to construct measures with certain desirable properties first on the real line

More information

Near supplements and complements in solvable groups of finite rank

Near supplements and complements in solvable groups of finite rank Near supplements and complements in solvable groups of finite rank Karl Lorensen (speaker) Pennsylvania State University, Altoona College Peter Kropholler (coauthor) University of Southampton August 8,

More information

COUNTABLY COMPLEMENTABLE LINEAR ORDERINGS

COUNTABLY COMPLEMENTABLE LINEAR ORDERINGS COUNTABLY COMPLEMENTABLE LINEAR ORDERINGS July 4, 2006 ANTONIO MONTALBÁN Abstract. We say that a countable linear ordering L is countably complementable if there exists a linear ordering L, possibly uncountable,

More information

Representing Scott Sets in Algebraic Settings

Representing Scott Sets in Algebraic Settings Representing Scott Sets in Algebraic Settings Alf Dolich Kingsborough Community College Julia F. Knight University of Notre Dame Karen Lange Wellesley College David Marker University of Illinois at Chicago

More information

Minimal non-p C-groups

Minimal non-p C-groups Algebra and Discrete Mathematics Volume 18 (2014). Number 1, pp. 1 7 Journal Algebra and Discrete Mathematics RESEARCH ARTICLE Minimal non-p C-groups Orest D. Artemovych Communicated by L. A. Kurdachenko

More information

Finitary Permutation Groups

Finitary Permutation Groups Finitary Permutation Groups Combinatorics Study Group Notes by Chris Pinnock You wonder and you wonder until you wander out into Infinity, where - if it is to be found anywhere - Truth really exists. Marita

More information

Jónsson posets and unary Jónsson algebras

Jónsson posets and unary Jónsson algebras Jónsson posets and unary Jónsson algebras Keith A. Kearnes and Greg Oman Abstract. We show that if P is an infinite poset whose proper order ideals have cardinality strictly less than P, and κ is a cardinal

More information

TORSION CLASSES AND PURE SUBGROUPS

TORSION CLASSES AND PURE SUBGROUPS PACIFIC JOURNAL OF MATHEMATICS Vol. 33 ; No. 1, 1970 TORSION CLASSES AND PURE SUBGROUPS B. J. GARDNER In this note we obtain a classification of the classes J?~ of abelian groups satisfying the following

More information

STRONGLY JÓNSSON AND STRONGLY HS MODULES

STRONGLY JÓNSSON AND STRONGLY HS MODULES STRONGLY JÓNSSON AND STRONGLY HS MODULES GREG OMAN Abstract. Let R be a commutative ring with identity and let M be an infinite unitary R-module. Then M is a Jónsson module provided every proper R-submodule

More information

Maximal non-commuting subsets of groups

Maximal non-commuting subsets of groups Maximal non-commuting subsets of groups Umut Işık March 29, 2005 Abstract Given a finite group G, we consider the problem of finding the maximal size nc(g) of subsets of G that have the property that no

More information

Algebra SEP Solutions

Algebra SEP Solutions Algebra SEP Solutions 17 July 2017 1. (January 2017 problem 1) For example: (a) G = Z/4Z, N = Z/2Z. More generally, G = Z/p n Z, N = Z/pZ, p any prime number, n 2. Also G = Z, N = nz for any n 2, since

More information

IN POSITIVE CHARACTERISTICS: 3. Modular varieties with Hecke symmetries. 7. Foliation and a conjecture of Oort

IN POSITIVE CHARACTERISTICS: 3. Modular varieties with Hecke symmetries. 7. Foliation and a conjecture of Oort FINE STRUCTURES OF MODULI SPACES IN POSITIVE CHARACTERISTICS: HECKE SYMMETRIES AND OORT FOLIATION 1. Elliptic curves and their moduli 2. Moduli of abelian varieties 3. Modular varieties with Hecke symmetries

More information

Prime Factorization and GCF. In my own words

Prime Factorization and GCF. In my own words Warm- up Problem What is a prime number? A PRIME number is an INTEGER greater than 1 with EXACTLY 2 positive factors, 1 and the number ITSELF. Examples of prime numbers: 2, 3, 5, 7 What is a composite

More information

GENERALIZED PIGEONHOLE PROPERTIES OF GRAPHS AND ORIENTED GRAPHS

GENERALIZED PIGEONHOLE PROPERTIES OF GRAPHS AND ORIENTED GRAPHS GENERALIZED PIGEONHOLE PROPERTIES OF GRAPHS AND ORIENTED GRAPHS ANTHONY BONATO, PETER CAMERON, DEJAN DELIĆ, AND STÉPHAN THOMASSÉ ABSTRACT. A relational structure A satisfies the n k property if whenever

More information

Topics in Module Theory

Topics in Module Theory Chapter 7 Topics in Module Theory This chapter will be concerned with collecting a number of results and constructions concerning modules over (primarily) noncommutative rings that will be needed to study

More information

ON VARIETIES IN WHICH SOLUBLE GROUPS ARE TORSION-BY-NILPOTENT

ON VARIETIES IN WHICH SOLUBLE GROUPS ARE TORSION-BY-NILPOTENT ON VARIETIES IN WHICH SOLUBLE GROUPS ARE TORSION-BY-NILPOTENT GÉRARD ENDIMIONI C.M.I., Université de Provence, UMR-CNRS 6632 39, rue F. Joliot-Curie, 13453 Marseille Cedex 13, France E-mail: endimion@gyptis.univ-mrs.fr

More information

FINITELY-GENERATED ABELIAN GROUPS

FINITELY-GENERATED ABELIAN GROUPS FINITELY-GENERATED ABELIAN GROUPS Structure Theorem for Finitely-Generated Abelian Groups Let G be a finitely-generated abelian group Then there exist a nonnegative integer t and (if t > 0) integers 1

More information

(a). W contains the zero vector in R n. (b). W is closed under addition. (c). W is closed under scalar multiplication.

(a). W contains the zero vector in R n. (b). W is closed under addition. (c). W is closed under scalar multiplication. . Subspaces of R n Bases and Linear Independence Definition. Subspaces of R n A subset W of R n is called a subspace of R n if it has the following properties: (a). W contains the zero vector in R n. (b).

More information

DUAL MODULES OVER A VALUATION RING. I

DUAL MODULES OVER A VALUATION RING. I DUAL MODULES OVER A VALUATION RING. I IRVING KAPLANSKY1 1. Introduction. The present investigation is inspired by a series of papers in the literature, beginning with Prüfer [8],2 and continuing with Pietrkowski

More information

Unipotent automorphisms of solvable groups

Unipotent automorphisms of solvable groups Department of Mathematical Sciences University of Bath Groups St Andrews 2017 in Birmingham 1. Introduction. 2. Solvable groups acting n-unipotently on solvable groups. 3. Examples. 1. Introduction 1.

More information

Modules over Non-Noetherian Domains

Modules over Non-Noetherian Domains Mathematical Surveys and Monographs Volume 84 Modules over Non-Noetherian Domains Laszlo Fuchs Luigi Sake American Mathematical Society Table of Contents Preface List of Symbols xi xv Chapter I. Commutative

More information

Linear Algebra. Paul Yiu. Department of Mathematics Florida Atlantic University. Fall 2011

Linear Algebra. Paul Yiu. Department of Mathematics Florida Atlantic University. Fall 2011 Linear Algebra Paul Yiu Department of Mathematics Florida Atlantic University Fall 2011 Linear Algebra Paul Yiu Department of Mathematics Florida Atlantic University Fall 2011 1A: Vector spaces Fields

More information

A BOREL SOLUTION TO THE HORN-TARSKI PROBLEM. MSC 2000: 03E05, 03E20, 06A10 Keywords: Chain Conditions, Boolean Algebras.

A BOREL SOLUTION TO THE HORN-TARSKI PROBLEM. MSC 2000: 03E05, 03E20, 06A10 Keywords: Chain Conditions, Boolean Algebras. A BOREL SOLUTION TO THE HORN-TARSKI PROBLEM STEVO TODORCEVIC Abstract. We describe a Borel poset satisfying the σ-finite chain condition but failing to satisfy the σ-bounded chain condition. MSC 2000:

More information

6. Dynkin quivers, Euclidean quivers, wild quivers.

6. Dynkin quivers, Euclidean quivers, wild quivers. 6 Dynkin quivers, Euclidean quivers, wild quivers This last section is more sketchy, its aim is, on the one hand, to provide a short survey concerning the difference between the Dynkin quivers, the Euclidean

More information

Very Flat, Locally Very Flat, and Contraadjusted Modules

Very Flat, Locally Very Flat, and Contraadjusted Modules Very Flat, Locally Very Flat, and Contraadjusted Modules Charles University in Prague, Faculty of Mathematics and Physics 27 th April 2016 Introducing the classes Throughout the whole talk R = commutative

More information

LIMIT GROUPS FOR RELATIVELY HYPERBOLIC GROUPS, II: MAKANIN-RAZBOROV DIAGRAMS

LIMIT GROUPS FOR RELATIVELY HYPERBOLIC GROUPS, II: MAKANIN-RAZBOROV DIAGRAMS LIMIT GROUPS FOR RELATIVELY HYPERBOLIC GROUPS, II: MAKANIN-RAZBOROV DIAGRAMS DANIEL GROVES Abstract. Let Γ be a torsion-free group which is hyperbolic relative to a collection of free abeian subgroups.

More information

Math 594. Solutions 5

Math 594. Solutions 5 Math 594. Solutions 5 Book problems 6.1: 7. Prove that subgroups and quotient groups of nilpotent groups are nilpotent (your proof should work for infinite groups). Give an example of a group G which possesses

More information

PETER A. CHOLAK, PETER GERDES, AND KAREN LANGE

PETER A. CHOLAK, PETER GERDES, AND KAREN LANGE D-MAXIMAL SETS PETER A. CHOLAK, PETER GERDES, AND KAREN LANGE Abstract. Soare [20] proved that the maximal sets form an orbit in E. We consider here D-maximal sets, generalizations of maximal sets introduced

More information

Primal, completely irreducible, and primary meet decompositions in modules

Primal, completely irreducible, and primary meet decompositions in modules Bull. Math. Soc. Sci. Math. Roumanie Tome 54(102) No. 4, 2011, 297 311 Primal, completely irreducible, and primary meet decompositions in modules by Toma Albu and Patrick F. Smith Abstract This paper was

More information

Notes on p-divisible Groups

Notes on p-divisible Groups Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete

More information