Maximal compact subgroups in the o-minimal setting

... We denote by G0 the definably connected component of the identity in a definable group G. It is the smallest definable subgroup of finite index in G ([21]). We say that a group G is definably connected if it is definable and G = G0 , so if G has no proper definable subgroup of finite index. A defina ...

... We denote by G0 the definably connected component of the identity in a definable group G. It is the smallest definable subgroup of finite index in G ([21]). We say that a group G is definably connected if it is definable and G = G0 , so if G has no proper definable subgroup of finite index. A defina ...

Coarse Structures on Infinite Groups

... (U1) if E ∈ E, then E −1 := {(y , x) ∈ X × X | (x, y ) ∈ E } ∈ E; (U2) if E , F ∈ E, then E ◦F := {(x, y ) ∈ X ×X | ∃z ∈ X s.t. (x, z) ∈ E , (z, y ) ∈ F }. (I1) and (I2) say that E is an ideal of X × X . Replacing (I1) and (I2) with their dual properties (F1) and (F2) (saying that E is a filter of X ...

... (U1) if E ∈ E, then E −1 := {(y , x) ∈ X × X | (x, y ) ∈ E } ∈ E; (U2) if E , F ∈ E, then E ◦F := {(x, y ) ∈ X ×X | ∃z ∈ X s.t. (x, z) ∈ E , (z, y ) ∈ F }. (I1) and (I2) say that E is an ideal of X × X . Replacing (I1) and (I2) with their dual properties (F1) and (F2) (saying that E is a filter of X ...

An Introduction to Topological Groups

... Since G → G, x 7→ yx is a homeomorphism, there exists a neighborhood W of e such that yW ⊆ V . Observe, that yW is a neighborhood of y ! By the continuity of the multiplication at (e, e), there is an open neighborhood U of e such that U · U ⊆ W . W.l.o.g. we may assume that U is symmetric (i.e. U = ...

... Since G → G, x 7→ yx is a homeomorphism, there exists a neighborhood W of e such that yW ⊆ V . Observe, that yW is a neighborhood of y ! By the continuity of the multiplication at (e, e), there is an open neighborhood U of e such that U · U ⊆ W . W.l.o.g. we may assume that U is symmetric (i.e. U = ...

Group Theory: The Journey Continues (Part I) (PDF) (296 KB, 27 pages)

... study of groups so exciting and important. So far, we have acquired a fairly thorough understanding of why groups are defined the way they are. We are now in a position to reap the benefits of this appreciation and explore the inner terrain of groups. Due to size and logical breakup considerations, ...

... study of groups so exciting and important. So far, we have acquired a fairly thorough understanding of why groups are defined the way they are. We are now in a position to reap the benefits of this appreciation and explore the inner terrain of groups. Due to size and logical breakup considerations, ...

COARSE GEOMETRY OF TOPOLOGICAL GROUPS Contents 1

... though the basic tools have been available for quite some time. Indeed, by a result of R. Struble [35] dating back to 1951, every locally compact second countable group admits a compatible left-invariant proper metric, i.e., so that the closed balls are compact. Struble’s theorem was based on an ear ...

... though the basic tools have been available for quite some time. Indeed, by a result of R. Struble [35] dating back to 1951, every locally compact second countable group admits a compatible left-invariant proper metric, i.e., so that the closed balls are compact. Struble’s theorem was based on an ear ...

Group Theory

... Most lectures on group theory actually start with the definition of what is a group. It may be worth though spending a few lines to mention how mathematicians came up with such a concept. Around 1770, Lagrange initiated the study of permutations in connection with the study of the solution of equati ...

... Most lectures on group theory actually start with the definition of what is a group. It may be worth though spending a few lines to mention how mathematicians came up with such a concept. Around 1770, Lagrange initiated the study of permutations in connection with the study of the solution of equati ...

M14/13

... We will call associative dihedral-like automorphic loops dihedral-like groups. The dihedrallike groups encompass the dihedral groups Dih(2, Zn , 1) = D2n , the generalized dihedral groups Dih(2, G, 1) = Dih(G), and certain generalized dicyclic groups Dih(4, G, 1). Recall that for an abelian group A ...

... We will call associative dihedral-like automorphic loops dihedral-like groups. The dihedrallike groups encompass the dihedral groups Dih(2, Zn , 1) = D2n , the generalized dihedral groups Dih(2, G, 1) = Dih(G), and certain generalized dicyclic groups Dih(4, G, 1). Recall that for an abelian group A ...

A REDUCTION TO THE COMPACT CASE FOR GROUPS

... The subgroup H can be decomposed as a product of definably connected subgroups H = A · N , with A abelian and N nilpotent. The previous theorem and the work on torsion-free definable groups made in [PeSta], yields a reduction to the compact case in analogy to the real case Fact 1.1: Theorem 1.3. Eve ...

... The subgroup H can be decomposed as a product of definably connected subgroups H = A · N , with A abelian and N nilpotent. The previous theorem and the work on torsion-free definable groups made in [PeSta], yields a reduction to the compact case in analogy to the real case Fact 1.1: Theorem 1.3. Eve ...

18 Divisible groups

... on f, g is equivalent to saying that ∆(A ∩ B) lies in the kernel of f ⊕ g : A ⊕ B → G. Consequently, we get an induced map on the quotient: f +g : A+B → G. Theorem 18.6. A group G is divisible if and only if it satisfies the following “injectivity” condition: Any homomorphism f : A → G from any grou ...

... on f, g is equivalent to saying that ∆(A ∩ B) lies in the kernel of f ⊕ g : A ⊕ B → G. Consequently, we get an induced map on the quotient: f +g : A+B → G. Theorem 18.6. A group G is divisible if and only if it satisfies the following “injectivity” condition: Any homomorphism f : A → G from any grou ...

Computable Completely Decomposable Groups.

... closure as be ones where any other computable algebraic closure was computably isomorphic to it. Quite aside from the basic natural interest in effective procedures in algebraic structures, we remark that computable structure theory often reveals deeper algebraic facts about familiar structures. For ...

... closure as be ones where any other computable algebraic closure was computably isomorphic to it. Quite aside from the basic natural interest in effective procedures in algebraic structures, we remark that computable structure theory often reveals deeper algebraic facts about familiar structures. For ...

670 notes - OSU Department of Mathematics

... Proof. One direction is easy. In the other direction, suppose that k and n are relatively prime. Multiply each element of (Z/nZ) by the class of k; argue that the resulting elements are all distinct. Thus one of them is the element 1. Another proof of the hard direction. (This proof provides a prac ...

... Proof. One direction is easy. In the other direction, suppose that k and n are relatively prime. Multiply each element of (Z/nZ) by the class of k; argue that the resulting elements are all distinct. Thus one of them is the element 1. Another proof of the hard direction. (This proof provides a prac ...

Automorphism groups of cyclic codes Rolf Bienert · Benjamin Klopsch

... disappear in the context of binary linear codes; cf. [3, Section 1.5].) It can be shown that every finite group arises as the automorphism group of a suitable binary linear code; cf. [9]. The question which finite permutation groups, i.e. finite groups with a fixed faithful permutation representatio ...

... disappear in the context of binary linear codes; cf. [3, Section 1.5].) It can be shown that every finite group arises as the automorphism group of a suitable binary linear code; cf. [9]. The question which finite permutation groups, i.e. finite groups with a fixed faithful permutation representatio ...

Some topics in the theory of finite groups

... at 2014 PhD Summer School in Discrete Mathematics and SYGN, Rogla, Slovenia. Since the summer school was aimed primarily at PhD students who are working in the latter area and may not necessarily be experts in group theory, the notes give a fairly general introduction to three main topics: Finite Si ...

... at 2014 PhD Summer School in Discrete Mathematics and SYGN, Rogla, Slovenia. Since the summer school was aimed primarily at PhD students who are working in the latter area and may not necessarily be experts in group theory, the notes give a fairly general introduction to three main topics: Finite Si ...

On finite primary rings and their groups of units

... cyclic. For if 0 Nk+1 Nk, Nk is a vector space over the field R/N and thus Nk cannot be cyclic. But Nk ~ 1+Nk, a subgroup of the group G of units of R. Hence G cannot be cyclic if N ~ 0. If N 0, R is a field and it is easy to see that its non-zero elements do not form an (infinité) cyclic group. If ...

... cyclic. For if 0 Nk+1 Nk, Nk is a vector space over the field R/N and thus Nk cannot be cyclic. But Nk ~ 1+Nk, a subgroup of the group G of units of R. Hence G cannot be cyclic if N ~ 0. If N 0, R is a field and it is easy to see that its non-zero elements do not form an (infinité) cyclic group. If ...

groups with exponent six - (DIMACS) Rutgers

... with B(2, 6) is a challenging problem. It seems unlikely that the computer-based methods now available — coset enumeration and Knuth-Bendix string rewriting — will succeed either in proving B(2, 6) itself is finite or in finding ‘small’ defining sets in the foreseeable future because the order of B( ...

... with B(2, 6) is a challenging problem. It seems unlikely that the computer-based methods now available — coset enumeration and Knuth-Bendix string rewriting — will succeed either in proving B(2, 6) itself is finite or in finding ‘small’ defining sets in the foreseeable future because the order of B( ...

Subgroups

... Therefore, a−1 ∈ Z(G). Hence, Z(G) is a subgroup of G. The union of subgroups is not necessarily a subgroup, but the intersection of subgroups is always a subgroup. Before I prove this, a word about notation. In this result, I want to talk about a bunch of subgroups of a group G. How should I denote ...

... Therefore, a−1 ∈ Z(G). Hence, Z(G) is a subgroup of G. The union of subgroups is not necessarily a subgroup, but the intersection of subgroups is always a subgroup. Before I prove this, a word about notation. In this result, I want to talk about a bunch of subgroups of a group G. How should I denote ...

MATH 436 Notes: Finitely generated Abelian groups.

... and so |Y | < ∞ and 2|Y | = 2|X| which yields |X| = |Y | as desired. So it only remains to consider the case |X| = |Y | = ∞. Let Pf inite (X) = {S ⊂ X||S| < ∞} be the set of finite subsets of X. Then there is a bijection Θ : Pf inite (X) → ⊕x∈X Z/2Z given by Θ(S) = (nx )x∈X where nx ∈ Z/2Z and nx = ...

... and so |Y | < ∞ and 2|Y | = 2|X| which yields |X| = |Y | as desired. So it only remains to consider the case |X| = |Y | = ∞. Let Pf inite (X) = {S ⊂ X||S| < ∞} be the set of finite subsets of X. Then there is a bijection Θ : Pf inite (X) → ⊕x∈X Z/2Z given by Θ(S) = (nx )x∈X where nx ∈ Z/2Z and nx = ...

Finitely generated groups with automatic presentations

... automata to check multiplication of words in L by generators. This concept was introduced in [6], motivated by work in hyperbolic manifolds as well as a general interest in computing on groups. The considerable success of the theory of automatic groups gives one motivation to have a general notion o ...

... automata to check multiplication of words in L by generators. This concept was introduced in [6], motivated by work in hyperbolic manifolds as well as a general interest in computing on groups. The considerable success of the theory of automatic groups gives one motivation to have a general notion o ...

Rewriting Systems for Coxeter Groups

... In all four cases, put a total ordering on the words in the alphabet S = {a, b, c} using a length-plus-lexicographic ordering with the ordering a > b > c on S. With these orderings, all of the rules v → w in cases I-IV satisfy v > w; since for any word v the number of words w with v > w is finite, ...

... In all four cases, put a total ordering on the words in the alphabet S = {a, b, c} using a length-plus-lexicographic ordering with the ordering a > b > c on S. With these orderings, all of the rules v → w in cases I-IV satisfy v > w; since for any word v the number of words w with v > w is finite, ...

Lecture 10 More on quotient groups

... The fibers of a homomorphism of groups can form their own group. In fact, the precise range of the homomorphism can be forgotten. We’re figuring out how to use the coset structure of those fibers to motivate forgetting the homomorphism as well (i.e. we’re looking for an internal criterion on subgrou ...

... The fibers of a homomorphism of groups can form their own group. In fact, the precise range of the homomorphism can be forgotten. We’re figuring out how to use the coset structure of those fibers to motivate forgetting the homomorphism as well (i.e. we’re looking for an internal criterion on subgrou ...

A group homomorphism is a function between two groups that links

... Let’s start with a familiar example, (Z/nZ, +). We can think of this as being constructed from the group (Z, +) and the subgroup nZ = {ns : s ∈ Z} as follows: We regard elements of Z (i.e., integers) as “equivalent” if their difference is in nZ (i.e., if they are congruent (mod n)). Then we form a n ...

... Let’s start with a familiar example, (Z/nZ, +). We can think of this as being constructed from the group (Z, +) and the subgroup nZ = {ns : s ∈ Z} as follows: We regard elements of Z (i.e., integers) as “equivalent” if their difference is in nZ (i.e., if they are congruent (mod n)). Then we form a n ...

LINEAR REPRESENTATIONS OF SOLUBLE GROUPS OF FINITE

... A consequence of their conjugacy is the following covering property: if G is a connected soluble group of finite Morley rank and N a definable, normal subgroup such that G/N is nilpotent, then G = N C where C is any Carter subgroup of G. The following lemma is an easy consequence of some work of Fre ...

... A consequence of their conjugacy is the following covering property: if G is a connected soluble group of finite Morley rank and N a definable, normal subgroup such that G/N is nilpotent, then G = N C where C is any Carter subgroup of G. The following lemma is an easy consequence of some work of Fre ...

Algebra I: Section 3. Group Theory 3.1 Groups.

... A−1 A = I = AA−1 . Nevertheless, certain subsets of M(n, F) are groups of great importance in geometry. To define them we must recall two facts from linear algebra. Theorem. An n × n matrix A has an inverse if and only if its determinant is nonzero: det A 6= 0. Moreover, there is an explicit algorit ...

... A−1 A = I = AA−1 . Nevertheless, certain subsets of M(n, F) are groups of great importance in geometry. To define them we must recall two facts from linear algebra. Theorem. An n × n matrix A has an inverse if and only if its determinant is nonzero: det A 6= 0. Moreover, there is an explicit algorit ...

more on the properties of almost connected pro-lie groups

... A projective limit of finite-dimensional Lie groups is called a pro-Lie group. [Lie group will always mean a finite-dimensional real Lie group.] In other words, a topological group G is a pro-Lie group if it is topologically isomorphic to a closed subgroup of an arbitrary product of Lie groups. Equi ...

... A projective limit of finite-dimensional Lie groups is called a pro-Lie group. [Lie group will always mean a finite-dimensional real Lie group.] In other words, a topological group G is a pro-Lie group if it is topologically isomorphic to a closed subgroup of an arbitrary product of Lie groups. Equi ...