DOCSLIB.ORG

MATH 413 – ADDITIONAL TOPICS in GROUP THEORY 1. Order in Abelian Groups 1.1. Order of a Product in an Abelian Group. the First

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
MATH 413 – ADDITIONAL TOPICS in GROUP THEORY 1. Order in Abelian Groups 1.1. Order of a Product in an Abelian Group. the First MATH 413 { ADDITIONAL TOPICS IN GROUP THEORY ALLAN YASHINSKI 1. Order in Abelian Groups 1.1. Order of a product in an abelian group. The first issue we shall address is the order of a product of two elements of finite order. Suppose G is a group and a; b 2 G have orders m = jaj and n = jbj. What can be said about jabj? Let's consider some abelian examples first. The following lemma will be used throughout. Lemma 1.1. Let G be an abelian group and a; b 2 G. Then for any n 2 Z, (ab)n = anbn: Exercise 1.A. Prove Lemma 1.1. (Hint: Prove it for n = 0 first. Then handle the case where n > 0 using induction. Lastly prove the case where n < 0 by using inverses to reduce to the case where n > 0.) Example 1. Let G = Z3 × Z4, a = (1; 0), and b = (0; 1). Then m = jaj = 3 and n = jbj = 4. Notice that ja + bj = j(1; 1)j = 12. Example 2. Let G = Z2 × Z4, a = (1; 0), and b = (0; 1). Then m = jaj = 2 and n = jbj = 4. Notice that ja + bj = j(1; 1)j = 4. Example 3. Let G = Z6 × Z4, a = (1; 0), and b = (0; 1). Then m = jaj = 6 and n = jbj = 4. Notice that ja + bj = j(1; 1)j = 12. Here is the general fact uniting these examples. Exercise 1.B. Let G = Zm × Zn and let a = (1; 0), b = (0; 1), so that m = jaj and n = jbj. Prove that ja + bj = lcm(m; n), the least common multiple1 of m and n. One may start to conjecture that if jaj = m and jbj = n, then jabj = lcm(m; n). However, this need not be the case, as the following examples show. Example 4. Suppose G is any group and a 2 G has order m. Then a−1 has order m also, and the product aa−1 = e has order 1, which is not equal to lcm(m; m) = m. Example 5. In Z12, consider a = 2 and b = 4. Then jaj = 6 and jbj = 3, and lcm(6; 3) = 6. However a + b = 6 has order 2 6= 6. Now we'll prove a result that is consistent with all of the above. Proposition 1.2. Let G be an abelian group and let a; b 2 G. Let m = jaj; n = jbj; and r = jabj. Then r j lcm(m; n). 1That is, lcm(m; n) is the smallest positive integer t such that m j t and n j t. It is a fact that mn lcm(m; n) = : gcd(m; n) 1 2 ALLAN YASHINSKI Proof. Let t = lcm(m; n). Since m j t and n j t, we have t = mk and t = n` for integers k; `. Using Lemma 1.1, (ab)t = atbt = amkbn` = (am)k(bn)` = ekb` = e: From Theorem 7.9 of the text, this implies r j t, as desired. This narrows down the possibilities for the order of a product ab in an abelian group. The answer will depend on more than just the numbers m = jaj and n = jbj. It will also depend on where a and b sit in the group in relation to each other2. We would like to identify situations where the order of ab is maximal, that is, equal to the least common multiple of jaj and jbj. Theorem 1.3. Let a; b be elements of an abelian group G and let m = jaj and n = jbj. If hai \ hbi = feg, then jabj = lcm(m; n). Proof. Let r = jabj and t = lcm(m; n). From Proposition 1.2, we have r j t, so r ≤ t. By definition of order, we have e = (ab)r = arbr, using Lemma 1.1. It follows that ar = b−r 2 hbi. Thus ar 2 hai \ hbi = feg, and so ar = e. Hence the order of a, which is m, must divide r. We also have br = (b−r)−1 = (ar)−1 = e−1 = e: So n j r as well. Hence r is a positive common multiple of m and n. By definition of least common multiple, t = lcm(m; n) ≤ r: We've shown r ≤ t and t ≤ r, so we've shown r = t. Compare the result of Theorem 1.3 with Example 4 and Example 5. In those examples, hai \ hbi 6= feg. Corollary 1.4. Let a; b be elements of an abelian group G and let m = jaj and n = jbj. If m and n are relatively prime, then jabj = mn. Exercise 1.C. Use Theorem 1.3 to prove Corollary 1.4. (Hint: hai \ hbi is a subgroup of both hai and hbi. Apply Lagrange's theorem to determine the order of hai \ hbi.) 1.2. Order of a product in nonabelian groups. For nonabelian groups, basi- cally nothing can be said about how to relate jabj to jaj and jbj. Example 6. Let G = S3, and consider a = (12) and b = (23). Then jaj = jbj = 2. However ab = (12)(23) = (123) is an element of order 3. Notice 3 has seemingly nothing to do with lcm(2; 2) = 2. Exercise 1.D. Consider the elements 0 −1 0 1 a = ; b = 1 0 −1 −1 of the nonabelian group GL(2; Z). Show that jaj = 4, jbj = 3, and yet ab has infinite order. 2A vague statement indeed. For example, if there is overlap between the cyclic subgroups hai and hbi, as in some of the above examples, the order of ab turns out to be less than lcm(jaj; jbj). MATH 413 { ADDITIONAL TOPICS IN GROUP THEORY 3 These examples stand in stark contrast to our above results for abelian groups. It gets much worse. Here is a somewhat difficult theorem, which we shall neither use nor prove3. Theorem 1.5. For any integers m; n; r > 1, there exists a finite group G and elements a; b 2 G such that jaj = m, jbj = n, and jabj = r. This theorem says that there is no relationship between jaj; jbj; and jabj in general nonabelian groups. Thus we should appreciate the results we have above for abelian groups. 1.3. Maximal order in finite abelian groups. We return to studying abelian groups. Let G be a finite abelian group. Since every element of G has finite order, it makes sense to discuss the largest order M of an element of G. Notice that M divides jGj by Lagrange's theorem, so M ≤ jGj. We also have M = jGj if and only if G is cyclic. Thus for non-cyclic abelian groups, M < jGj. We shall prove the following theorem. Theorem 1.6. Let G be a finite abelian group, and let M be the largest order of an element of G. Then for any a 2 G, the order of a divides M. In particular, aM = e for all a 2 G. Since M ≤ jGj, this strengthens the results on orders of elements obtained from Lagrange's theorem. Let's consider some examples before proving the theorem. Example 7. If G is cyclic, then G contains an element of order jGj by definition of cyclic. So M = jGj. In this case, the results of Theorem 1.6 coincide with Corollary 8.6 from the text. Example 8. Consider the abelian group G = Z4 × Z6. Notice that jGj = 24. The element (1; 1) has order 12, and this is the maximal order of an element of G because G is not cyclic. So M = 12 < 24 = jGj. Theorem 1.6 says that the order of every element of G divides 12, hence is 1; 2; 3; 4; 6; or 12. Further every (k; `) 2 Z4 × Z6 satisfies 12(k; `) = (0; 0). Example 9. The last example can be generalized to Zm × Zn. We claim that the maximal order M = lcm(m; n). Let t = lcm(m; n). Since m j t and n j t, we have t = mr and t = ns for some r; s 2 Z. Notice that for any (k; `) 2 Zm × Zn, t(k; `) = (tk; t`) = (mrk; ns`) = (0; 0) in Zm × Zn: By definition of order, we have j(k; `)j ≤ t. Since (k; `) 2 Zm × Zn was arbitrary and M is the order of some element of Zm × Zn, we see M ≤ t. On the other hand, the element (1; 1) has order t by Exercise 1.B, which shows M ≥ t. Thus M = t = lcm(m; n). In particular, we see Zm × Zn is cyclic if and only if lcm(m; n) = mn, which happens if and only if gcd(m; n) = 1. Example 10. One can consider products of cyclic groups with more factors. For example, the maximal order of an element of Z2 × Z2 × Z2 × Z2 is M = 2. The maximal order of an element of Z2 × Z3 × Z6 × Z8 is M = 24. As we shall see later, every finite abelian group is a product of cyclic groups. So these types of examples are the only examples to consider. Theorem 1.6 really is an abelian theorem. It fails badly for nonabelian groups. 3A proof can be found of page 28 of http://www.jmilne.org/math/CourseNotes/GT.pdf 4 ALLAN YASHINSKI Example 11. The nonabelian group G = S3 contains elements of order 1; 2; and 3 3. The largest order is 3, but 2 - 3, and certainly a does not hold for all a 2 S3. We will prove Theorem 1.6 using the following lemmas.
Load more

Recommended publications
  • Classification of Finite Abelian Groups
    Math 317 C1 John Sullivan Spring 2003 Classification of Finite Abelian Groups (Notes based on an article by Navarro in the Amer. Math. Monthly, February 2003.) The fundamental theorem of finite abelian groups expresses any such group as a product of cyclic groups: Theorem. Suppose G is a finite abelian group. Then G is (in a unique way) a direct product of cyclic groups of order pk with p prime. Our first step will be a special case of Cauchy’s Theorem, which we will prove later for arbitrary groups: whenever p |G| then G has an element of order p. Theorem (Cauchy). If G is a finite group, and p |G| is a prime, then G has an element of order p (or, equivalently, a subgroup of order p). ∼ Proof when G is abelian. First note that if |G| is prime, then G = Zp and we are done. In general, we work by induction. If G has no nontrivial proper subgroups, it must be a prime cyclic group, the case we’ve already handled. So we can suppose there is a nontrivial subgroup H smaller than G. Either p |H| or p |G/H|. In the first case, by induction, H has an element of order p which is also order p in G so we’re done. In the second case, if ∼ g + H has order p in G/H then |g + H| |g|, so hgi = Zkp for some k, and then kg ∈ G has order p. Note that we write our abelian groups additively. Definition. Given a prime p, a p-group is a group in which every element has order pk for some k.
    [Show full text]
  • Janko's Sporadic Simple Groups
    Janko’s Sporadic Simple Groups: a bit of history Algebra, Geometry and Computation CARMA, Workshop on Mathematics and Computation Terry Gagen and Don Taylor The University of Sydney 20 June 2015 Fifty years ago: the discovery In January 1965, a surprising announcement was communicated to the international mathematical community. Zvonimir Janko, working as a Research Fellow at the Institute of Advanced Study within the Australian National University had constructed a new sporadic simple group. Before 1965 only five sporadic simple groups were known. They had been discovered almost exactly one hundred years prior (1861 and 1873) by Émile Mathieu but the proof of their simplicity was only obtained in 1900 by G. A. Miller. Finite simple groups: earliest examples É The cyclic groups Zp of prime order and the alternating groups Alt(n) of even permutations of n 5 items were the earliest simple groups to be studied (Gauss,≥ Euler, Abel, etc.) É Evariste Galois knew about PSL(2,p) and wrote about them in his letter to Chevalier in 1832 on the night before the duel. É Camille Jordan (Traité des substitutions et des équations algébriques,1870) wrote about linear groups defined over finite fields of prime order and determined their composition factors. The ‘groupes abéliens’ of Jordan are now called symplectic groups and his ‘groupes hypoabéliens’ are orthogonal groups in characteristic 2. É Émile Mathieu introduced the five groups M11, M12, M22, M23 and M24 in 1861 and 1873. The classical groups, G2 and E6 É In his PhD thesis Leonard Eugene Dickson extended Jordan’s work to linear groups over all finite fields and included the unitary groups.
    [Show full text]
  • Orders on Computable Torsion-Free Abelian Groups
    Orders on Computable Torsion-Free Abelian Groups Asher M. Kach (Joint Work with Karen Lange and Reed Solomon) University of Chicago 12th Asian Logic Conference Victoria University of Wellington December 2011 Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 1 / 24 Outline 1 Classical Algebra Background 2 Computing a Basis 3 Computing an Order With A Basis Without A Basis 4 Open Questions Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 2 / 24 Torsion-Free Abelian Groups Remark Disclaimer: Hereout, the word group will always refer to a countable torsion-free abelian group. The words computable group will always refer to a (fixed) computable presentation. Definition A group G = (G : +; 0) is torsion-free if non-zero multiples of non-zero elements are non-zero, i.e., if (8x 2 G)(8n 2 !)[x 6= 0 ^ n 6= 0 =) nx 6= 0] : Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 3 / 24 Rank Theorem A countable abelian group is torsion-free if and only if it is a subgroup ! of Q . Definition The rank of a countable torsion-free abelian group G is the least κ cardinal κ such that G is a subgroup of Q . Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 4 / 24 Example The subgroup H of Q ⊕ Q (viewed as having generators b1 and b2) b1+b2 generated by b1, b2, and 2 b1+b2 So elements of H look like β1b1 + β2b2 + α 2 for β1; β2; α 2 Z.
    [Show full text]
  • Dihedral Groups Ii
    DIHEDRAL GROUPS II KEITH CONRAD We will characterize dihedral groups in terms of generators and relations, and describe the subgroups of Dn, including the normal subgroups. We will also introduce an infinite group that resembles the dihedral groups and has all of them as quotient groups. 1. Abstract characterization of Dn −1 −1 The group Dn has two generators r and s with orders n and 2 such that srs = r . We will show every group with a pair of generators having properties similar to r and s admits a homomorphism onto it from Dn, and is isomorphic to Dn if it has the same size as Dn. Theorem 1.1. Let G be generated by elements x and y where xn = 1 for some n ≥ 3, 2 −1 −1 y = 1, and yxy = x . There is a surjective homomorphism Dn ! G, and if G has order 2n then this homomorphism is an isomorphism. The hypotheses xn = 1 and y2 = 1 do not mean x has order n and y has order 2, but only that their orders divide n and divide 2. For instance, the trivial group has the form hx; yi where xn = 1, y2 = 1, and yxy−1 = x−1 (take x and y to be the identity). Proof. The equation yxy−1 = x−1 implies yxjy−1 = x−j for all j 2 Z (raise both sides to the jth power). Since y2 = 1, we have for all k 2 Z k ykxjy−k = x(−1) j by considering even and odd k separately. Thus k (1.1) ykxj = x(−1) jyk: This shows each product of x's and y's can have all the x's brought to the left and all the y's brought to the right.
    [Show full text]
  • Mathematics of the Rubik's Cube
    Mathematics of the Rubik's cube Associate Professor W. D. Joyner Spring Semester, 1996{7 2 \By and large it is uniformly true that in mathematics that there is a time lapse between a mathematical discovery and the moment it becomes useful; and that this lapse can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful." John von Neumann COLLECTED WORKS, VI, p. 489 For more mathematical quotes, see the first page of each chapter below, [M], [S] or the www page at http://math.furman.edu/~mwoodard/mquot. html 3 \There are some things which cannot be learned quickly, and time, which is all we have, must be paid heavily for their acquiring. They are the very simplest things, and because it takes a man's life to know them the little new that each man gets from life is very costly and the only heritage he has to leave." Ernest Hemingway (From A. E. Hotchner, PAPA HEMMINGWAY, Random House, NY, 1966) 4 Contents 0 Introduction 13 1 Logic and sets 15 1.1 Logic................................ 15 1.1.1 Expressing an everyday sentence symbolically..... 18 1.2 Sets................................ 19 2 Functions, matrices, relations and counting 23 2.1 Functions............................. 23 2.2 Functions on vectors....................... 28 2.2.1 History........................... 28 2.2.2 3 × 3 matrices....................... 29 2.2.3 Matrix multiplication, inverses.............. 30 2.2.4 Muliplication and inverses...............
    [Show full text]
  • Lie Groups and Lie Algebras
    2 LIE GROUPS AND LIE ALGEBRAS 2.1 Lie groups The most general definition of a Lie group G is, that it is a differentiable manifold with group structure, such that the multiplication map G G G, (g,h) gh, and the inversion map G G, g g−1, are differentiable.× → But we shall7→ not need this concept in full generality.→ 7→ In order to avoid more elaborate differential geometry, we will restrict attention to matrix groups. Consider the set of all invertible n n matrices with entries in F, where F stands either for the real (R) or complex× (C) numbers. It is easily verified to form a group which we denote by GL(n, F), called the general linear group in n dimensions over the number field F. The space of all n n matrices, including non- invertible ones, with entries in F is denoted by M(n,×F). It is an n2-dimensional 2 vector space over F, isomorphic to Fn . The determinant is a continuous function det : M(n, F) F and GL(n, F) = det−1(F 0 ), since a matrix is invertible → −{ } 2 iff its determinant is non zero. Hence GL(n, F) is an open subset of Fn . Group multiplication and inversion are just given by the corresponding familiar matrix 2 2 2 2 2 operations, which are differentiable functions Fn Fn Fn and Fn Fn respectively. × → → 2.1.1 Examples of Lie groups GL(n, F) is our main example of a (matrix) Lie group. Any Lie group that we encounter will be a subgroups of some GL(n, F).
    [Show full text]
  • Map Composition Generalized to Coherent Collections of Maps
    Front. Math. China 2015, 10(3): 547–565 DOI 10.1007/s11464-015-0435-5 Map composition generalized to coherent collections of maps Herng Yi CHENG1, Kang Hao CHEONG1,2 1 National University of Singapore High School of Mathematics and Science, Singapore 129957, Singapore 2 Tanglin Secondary School, Singapore 127391, Singapore c Higher Education Press and Springer-Verlag Berlin Heidelberg 2015 Abstract Relation algebras give rise to partial algebras on maps, which are generalized to partial algebras on polymaps while preserving the properties of relation union and composition. A polymap is defined as a map with every point in the domain associated with a special set of maps. Polymaps can be represented as small subcategories of Set∗, the category of pointed sets. Map composition and the counterpart of relation union for maps are generalized to polymap composition and sum. Algebraic structures and categories of polymaps are investigated. Polymaps present the unique perspective of an algebra that can retain many of its properties when its elements (maps) are augmented with collections of other elements. Keywords Relation algebra, partial algebra, composition MSC 08A02 1 Introduction This paper considers partial algebras on maps (similar to those defined in [15]), constructed in analogy to relation algebras [5,12,16]. The sum of two maps f0 : X0 → Y0 and f1 : X1 → Y1 is defined if the union of their graphs is a functional binary relation R ⊆ (X0 ∪ X1) × (Y0 ∪ Y1), in which case the sum is the map (f0 + f1): X0 ∪ X1 → Y0 ∪ Y1 with graph R. This is defined similarly to the union operation + from [10].
    [Show full text]
  • LOW-DIMENSIONAL UNITARY REPRESENTATIONS of B3 1. Introduction Unitary Braid Representations Have Been Constructed in Several
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 9, Pages 2597{2606 S 0002-9939(01)05903-2 Article electronically published on March 15, 2001 LOW-DIMENSIONAL UNITARY REPRESENTATIONS OF B3 IMRE TUBA (Communicated by Stephen D. Smith) Abstract. We characterize all simple unitarizable representations of the braid group B3 on complex vector spaces of dimension d ≤ 5. In particular, we prove that if σ1 and σ2 denote the two generating twists of B3,thenasimple representation ρ : B3 ! GL(V )(fordimV ≤ 5) is unitarizable if and only if the eigenvalues λ1,λ2;::: ;λd of ρ(σ1) are distinct, satisfy jλij =1and (d) ≤ ≤ (d) µ1i > 0for2 i d,wheretheµ1i are functions of the eigenvalues, explicitly described in this paper. 1. Introduction Unitary braid representations have been constructed in several ways using the representation theory of Kac-Moody algebras and quantum groups (see e.g. [1], [2], and [4]), and specializations of the reduced Burau and Gassner representations in [5]. Such representations easily lead to representations of PSL(2; Z)=B3=Z ,where Z is the center of B3,andPSL(2; Z)=SL(2; Z)={1g,where{1g is the center of SL(2; Z). We give a complete classification of simple unitary representations of B3 of dimension d ≤ 5 in this paper. In particular, the unitarizability of a braid rep- resentation depends only on the the eigenvalues λ1,λ2;::: ,λd of the images of the two generating twists of B3. The condition for unitarizability is a set of linear in- equalities in the logarithms of these eigenvalues. In other words, the representation is unitarizable if and only if the (arg λ1; arg λ2;::: ;arg λd) is a point inside a poly- hedron in (R=2π)d, where we give the equations of the hyperplanes that bound this polyhedron.
    [Show full text]
  • Countable Torsion-Free Abelian Groups
    COUNTABLE TORSION-FREE ABELIAN GROUPS By M. O'N. CAMPBELL [Received 27 January 1959.—Read 19 February 1959] Introduction IN this paper we present a new classification of the countable torsion-free abelian groups. Since any abelian group 0 may be regarded as an extension of a periodic group (namely the torsion part of G) by a torsion-free group, and since there exists the well-known complete classification of the count- able periodic abelian groups due to Ulm, it is clear that the classification problem studied here is of great interest. By a group we shall always mean an abelian group, and the additive notation will be employed throughout. It is well known that all the maximal linearly independent sets of elements of a given group are car- dinally equivalent; the corresponding cardinal number is the rank of the group. Derry (2) and Mal'cev (4) have studied the torsion-free groups of finite rank only, and have given classifications of these groups in terms of certain equivalence classes of systems of matrices with £>-adic elements. Szekeres (5) attempted to abolish the finiteness restriction on rank; he extracted certain invariants, but did not derive a complete classification. By a basis of a torsion-free group 0 we mean any maximal linearly independent subset of G. A subgroup generated by a basis of G will be called a basal subgroup or described as basal in G. Thus U is a basal sub- group of G if and only if (i) U is free abelian, and (ii) the factor group G/U is periodic.
    [Show full text]
  • Chapter 3: Transformations Groups, Orbits, and Spaces of Orbits
    Preprint typeset in JHEP style - HYPER VERSION Chapter 3: Transformations Groups, Orbits, And Spaces Of Orbits Gregory W. Moore Abstract: This chapter focuses on of group actions on spaces, group orbits, and spaces of orbits. Then we discuss mathematical symmetric objects of various kinds. May 3, 2019 -TOC- Contents 1. Introduction 2 2. Definitions and the stabilizer-orbit theorem 2 2.0.1 The stabilizer-orbit theorem 6 2.1 First examples 7 2.1.1 The Case Of 1 + 1 Dimensions 11 3. Action of a topological group on a topological space 14 3.1 Left and right group actions of G on itself 19 4. Spaces of orbits 20 4.1 Simple examples 21 4.2 Fundamental domains 22 4.3 Algebras and double cosets 28 4.4 Orbifolds 28 4.5 Examples of quotients which are not manifolds 29 4.6 When is the quotient of a manifold by an equivalence relation another man- ifold? 33 5. Isometry groups 34 6. Symmetries of regular objects 36 6.1 Symmetries of polygons in the plane 39 3 6.2 Symmetry groups of some regular solids in R 42 6.3 The symmetry group of a baseball 43 7. The symmetries of the platonic solids 44 7.1 The cube (\hexahedron") and octahedron 45 7.2 Tetrahedron 47 7.3 The icosahedron 48 7.4 No more regular polyhedra 50 7.5 Remarks on the platonic solids 50 7.5.1 Mathematics 51 7.5.2 History of Physics 51 7.5.3 Molecular physics 51 7.5.4 Condensed Matter Physics 52 7.5.5 Mathematical Physics 52 7.5.6 Biology 52 7.5.7 Human culture: Architecture, art, music and sports 53 7.6 Regular polytopes in higher dimensions 53 { 1 { 8.
    [Show full text]
  • Automorphisms of Direct Products of Groups and Their Geometric Realisations
    Math. Ann. 263, 343 364 ~1983) Am Springer-Verlag 1983 Automorphisms of Direct Products of Groups and Their Geometric Realisations Francis E. A. Johnson Department of Mathematics, University College, Gower Street, London WC1E 6BT, UK O. Introduction An outstanding problem in the theory of Poincar6 Duality (=PD) groups is to decide whether each PD group can be realised as the fundamental group of a smooth closed aspherical manifold. See [2, 15-17] for previous work on this problem. There are two main ways of constructing new PD groups from old [18] ; (i) by extension of one PD group by another, and (ii) by torsion free extension of a PD group by a finite group. One approach to the above problem is to apply the constructions (i) and (ii) to known realisable PD groups and try to decide whether the resulting PD groups are realisable. Though other examples are known, for instance the examples con- structed by Mostow and Siu in [25], the most convenient PD groups with which to start are discrete uniform subgroups of noncompact Lie groups. If F is a torsion free discrete uniform subgroup of a connected noncompact Lie group G, and if K is a maximal compact subgroup of G, then F\G/K is a smooth closed manifold of type K(F, 1). The present paper is a continuation of both [15] and [16]. Whereas in [16] we allowed only Surface groups, that is, torsion free discrete uniform subgroups of PGLz(IR), and in [15] allowed only discrete uniform subgroups of Lie groups with no PSLz(IR) factors as building blocks, here we go some way towards combining the two.
    [Show full text]
  • Finiteness of $ Z $-Classes in Reductive Groups
    FINITENESS OF z-CLASSES IN REDUCTIVE GROUPS SHRIPAD M. GARGE. AND ANUPAM SINGH. Abstract. Let k be a perfect field such that for every n there are only finitely many field extensions, up to isomorphism, of k of degree n. If G is a reductive algebraic group defined over k, whose characteristic is very good for G, then we prove that G(k) has only finitely many z-classes. For each perfect field k which does not have the above finiteness property we show that there exist groups G over k such that G(k) has infinitely many z-classes. 1. Introduction Let G be a group. A G-set is a non-empty set X admitting an action of the group G. Two G-sets X and Y are called G-isomorphic if there is a bijection φ : X → Y which commutes with the G-actions on X and Y . If the G-sets X and Y are transitive, say X = Gx and Y = Gy for some x ∈ X and y ∈ Y , then X and Y are G-isomorphic if and only if the stabilizers Gx and Gy of x and y, respectively, in G are conjugate as subgroups of G. One of the most important group actions is the conjugation action of a group G on itself. It partitions the group G into its conjugacy classes which are orbits under this action. However, from the point of view of G-action, it is more natural to partition the group G into sets consisting of conjugacy classes which are G- arXiv:2001.06359v1 [math.GR] 17 Jan 2020 isomorphic to each other.
    [Show full text]