The Type of the Classifying Space of a Topological Group for the
... Let G be totally disconnected and let F be the trivial family TR consisting of one element, namely the trivial group. We claim that then E(G, TR) is contractible if and only if G is discrete. If G is discrete, we already know that E(G, TR) is contractible. Suppose now that E(G, TR) is contractible. ...
... Let G be totally disconnected and let F be the trivial family TR consisting of one element, namely the trivial group. We claim that then E(G, TR) is contractible if and only if G is discrete. If G is discrete, we already know that E(G, TR) is contractible. Suppose now that E(G, TR) is contractible. ...
TERNARY BOOLEAN ALGEBRA 1. Introduction. The
... the latter [l, 2 ] , 2 and by the former we shall mean simply a function of three variables defined for elements of a set K whose values are also in K. Ternary operations have been discussed in groupoids [4] and groups [3 ] ; in Boolean algebra an operation different from the one introduced here was ...
... the latter [l, 2 ] , 2 and by the former we shall mean simply a function of three variables defined for elements of a set K whose values are also in K. Ternary operations have been discussed in groupoids [4] and groups [3 ] ; in Boolean algebra an operation different from the one introduced here was ...
Linear Congruences
... In ordinary algebra, an equation of the form ax = b (where a and b are given real numbers) is called a linear equation, and its solution x = b/a is obtained by multiplying both sides of the equation by a−1 = 1/a. The subject of this lecture is how to solve any linear congruence ax ≡ b (mod m) where ...
... In ordinary algebra, an equation of the form ax = b (where a and b are given real numbers) is called a linear equation, and its solution x = b/a is obtained by multiplying both sides of the equation by a−1 = 1/a. The subject of this lecture is how to solve any linear congruence ax ≡ b (mod m) where ...
separability, the countable chain condition and the lindelof property
... pEP and qEQ such that p^y^q. (2.2) Theorem. A LOTS satisfying the CCC is hereditarily Lindelof. Proof. Suppose that X is a LOTS satisfying the CCC. To show that X is hereditarily Lindelof, it will suffice to show that if V is any collection of open intervals in X, then there is a countable subcollec ...
... pEP and qEQ such that p^y^q. (2.2) Theorem. A LOTS satisfying the CCC is hereditarily Lindelof. Proof. Suppose that X is a LOTS satisfying the CCC. To show that X is hereditarily Lindelof, it will suffice to show that if V is any collection of open intervals in X, then there is a countable subcollec ...
On the Universal Space for Group Actions with Compact Isotropy
... for the family F is a G-CW -complex E(G, F) such that the fixed point set E(G, F)H is weakly contractible for H ∈ F and all its isotropy groups belong to F. Recall that a map f : X → Y of spaces is a weak homotopy equivalence if and only if the induced map f∗ : πn (X, x) → πn (Y, f (x)) is an isomo ...
... for the family F is a G-CW -complex E(G, F) such that the fixed point set E(G, F)H is weakly contractible for H ∈ F and all its isotropy groups belong to F. Recall that a map f : X → Y of spaces is a weak homotopy equivalence if and only if the induced map f∗ : πn (X, x) → πn (Y, f (x)) is an isomo ...
Isomorphisms - KSU Web Home
... Lemma 292 Let G be a group and let H = fTg j g 2 Gg where Tg is de…ned as in lemma 291. Then, H is a group with function composition. Proof. We have to verify that (H; ) satis…es the four properties of a group. Closure: Let Ta and Tb be two elements of H. Show Ta Tb belongs to H. An element of H is ...
... Lemma 292 Let G be a group and let H = fTg j g 2 Gg where Tg is de…ned as in lemma 291. Then, H is a group with function composition. Proof. We have to verify that (H; ) satis…es the four properties of a group. Closure: Let Ta and Tb be two elements of H. Show Ta Tb belongs to H. An element of H is ...
Let n be a positive integer. Recall that we say that integers a, b are
... To prove that two groups are isomorphic usually requires finding an explicit isomorphism. Proving that two groups are not isomorphic is often easier, as if we can find an “abstract property” that distinguishes them, then this is enough, since isomorphic groups have the same “abstract properties”. We ...
... To prove that two groups are isomorphic usually requires finding an explicit isomorphism. Proving that two groups are not isomorphic is often easier, as if we can find an “abstract property” that distinguishes them, then this is enough, since isomorphic groups have the same “abstract properties”. We ...
notes on the subspace theorem
... Roth’s Theorem is easy to prove if α ∈ C\R, or if α is a real quadratic number. For real algebraic numbers α of degree > 3, the proof of Roth’s Theorem is very difficult. In the formulation of the Subspace Theorem, we need some notions from linear algebra, which we recall below. Let n be an integer ...
... Roth’s Theorem is easy to prove if α ∈ C\R, or if α is a real quadratic number. For real algebraic numbers α of degree > 3, the proof of Roth’s Theorem is very difficult. In the formulation of the Subspace Theorem, we need some notions from linear algebra, which we recall below. Let n be an integer ...
(pdf)
... OK is finitely generated over Z–the result then being immediate from the structure theorem for finitely generated abelian groups. We will omit the proof of this, but it can be found on pages 12-13 of Neukirch. However, using this fact, we are able to prove the statement we set out to prove at the be ...
... OK is finitely generated over Z–the result then being immediate from the structure theorem for finitely generated abelian groups. We will omit the proof of this, but it can be found on pages 12-13 of Neukirch. However, using this fact, we are able to prove the statement we set out to prove at the be ...
1. Introduction 2. Examples and arithmetic of Boolean algebras
... with −Y the complement X\Y of Y with respect to X, is a Boolean algebra: the axioms (B1) through (B5) simply state elementary laws of set theory. P(X) is called the power set algebra of X. Definition 4. A subalgebra of a power set algebra P(X) is called an algebra of subsets of X or an algebra of se ...
... with −Y the complement X\Y of Y with respect to X, is a Boolean algebra: the axioms (B1) through (B5) simply state elementary laws of set theory. P(X) is called the power set algebra of X. Definition 4. A subalgebra of a power set algebra P(X) is called an algebra of subsets of X or an algebra of se ...
Trigonometric sums
... Thus, F (ψχ −1 ) is totally ramified at 0. At ∞, we know that F (χ −1 ) is tamely ramified. If ψ is trivial, then F (ψ) is non-ramified and hence, F (χ −1 ψ) is also totally tamely ramified. If ψ is non-trivial, then F (ψ) is totally wildly ramified (with Swan conductor 1). Thus, F (χ −1 ψ) is also ...
... Thus, F (ψχ −1 ) is totally ramified at 0. At ∞, we know that F (χ −1 ) is tamely ramified. If ψ is trivial, then F (ψ) is non-ramified and hence, F (χ −1 ψ) is also totally tamely ramified. If ψ is non-trivial, then F (ψ) is totally wildly ramified (with Swan conductor 1). Thus, F (χ −1 ψ) is also ...
Equations and Dot-Depth One By: Francine Blanchet
... of equations which characterize V1,1, V1,2 and V1,3 are equivalent to finite ones. Generalizations to V2,1 are discussed. (Knast [9,10] provide an equation system for level one of Brzozowski's dot-depth hierarchy.) The proofs rely on some combinatorial properties of the congruences ~(m) and ~(1,m) s ...
... of equations which characterize V1,1, V1,2 and V1,3 are equivalent to finite ones. Generalizations to V2,1 are discussed. (Knast [9,10] provide an equation system for level one of Brzozowski's dot-depth hierarchy.) The proofs rely on some combinatorial properties of the congruences ~(m) and ~(1,m) s ...
on the burnside problem on periodic groups
... In 1982 Ol'shanskii [10] succeeded in finding a considerably shorter proof of the theorem of Novikov and Adian, although the estimate n > 1010 (where n again is odd) of [10] is much worse than n > 665 of Adian's [1]. On the other hand, it is worth noting that the approach of Ol'shanskii's to treat t ...
... In 1982 Ol'shanskii [10] succeeded in finding a considerably shorter proof of the theorem of Novikov and Adian, although the estimate n > 1010 (where n again is odd) of [10] is much worse than n > 665 of Adian's [1]. On the other hand, it is worth noting that the approach of Ol'shanskii's to treat t ...
PERIODS OF GENERIC TORSORS OF GROUPS OF
... When Q is an algebraic torus, the period of a generic Q-torsor is studied in [7]. Based on the technique developed in [7], we generalize the results to the case where Q is a smooth group of multiplicative type (see Corollary 4.3). We also study in Section 5 the relationships among the periods of gen ...
... When Q is an algebraic torus, the period of a generic Q-torsor is studied in [7]. Based on the technique developed in [7], we generalize the results to the case where Q is a smooth group of multiplicative type (see Corollary 4.3). We also study in Section 5 the relationships among the periods of gen ...
![arXiv:math/0604168v1 [math.CO] 7 Apr 2006](http://s1.studyres.com/store/data/017890502_1-2c1abc75bd42752544cdf6d7b46b6ed7-300x300.png)
arXiv:math/0604168v1 [math.CO] 7 Apr 2006
... constructing a multiset of hyperplanes corresponding to the kernels of the linear forms defined by the columns, giving a multiarrangement, and disregarding the multiplicities to obtain an arrangement. If A is essential, the hyperplanes in A intersect in the zero subspace, and the matrix A has full r ...
... constructing a multiset of hyperplanes corresponding to the kernels of the linear forms defined by the columns, giving a multiarrangement, and disregarding the multiplicities to obtain an arrangement. If A is essential, the hyperplanes in A intersect in the zero subspace, and the matrix A has full r ...