y - Algebra House
... materials from this site and presentations, without the express written consent of Mr. Sims, is prohibited. © Mr. Sims. All rights reserved. ...
... materials from this site and presentations, without the express written consent of Mr. Sims, is prohibited. © Mr. Sims. All rights reserved. ...
Every H-decomposition of Kn has a nearly resolvable
... such that the following holds: If S is an huniform hypergraph with n>N0 vertices and: – There exists d > en such that for every ...
... such that the following holds: If S is an huniform hypergraph with n>N0 vertices and: – There exists d > en such that for every ...
Finite Abelian Groups as Galois Groups
... Proof. Since Q(ϵ) is the splitting field for X n −1 over Q and the extension is separable, the extension is Galois. We first show that [Q(ϵ) : Q] = ϕ(n). To see this, set G := Gal(Q(ϵ)/Q) and let σ ∈ G. Then σ is determined by its value σ(ϵ). On the one hand, its easy to check that σ(ϵ) is also a pr ...
... Proof. Since Q(ϵ) is the splitting field for X n −1 over Q and the extension is separable, the extension is Galois. We first show that [Q(ϵ) : Q] = ϕ(n). To see this, set G := Gal(Q(ϵ)/Q) and let σ ∈ G. Then σ is determined by its value σ(ϵ). On the one hand, its easy to check that σ(ϵ) is also a pr ...
Cosets, factor groups, direct products, homomorphisms, isomorphisms
... Some motivating words and thoughts of wisdom!} By Cayley’s theorem any whatever small or big group G can be found inside the symmetric group of all permutations on enough many elements and more specifically Cayley’s theorem states that can be always done inside S∣G∣ . So, for example a group of 8 el ...
... Some motivating words and thoughts of wisdom!} By Cayley’s theorem any whatever small or big group G can be found inside the symmetric group of all permutations on enough many elements and more specifically Cayley’s theorem states that can be always done inside S∣G∣ . So, for example a group of 8 el ...
Solutions
... 2. (25 pts.) Let V and W be vector spaces and let T : V → W be linear. If V1 is a vector subspace of V show that T(V1 ) = {w ∈ R(T) | T(v) = w for some v ∈ V1 } is a vector subspace of W. Solution: We must show that (a) 0W ∈ T(V1 ), (b) w1 + w2 ∈ T(V1 ), for all w1 , w2 ∈ T(V1 ), and (c) cw1 ∈ T(V1 ...
... 2. (25 pts.) Let V and W be vector spaces and let T : V → W be linear. If V1 is a vector subspace of V show that T(V1 ) = {w ∈ R(T) | T(v) = w for some v ∈ V1 } is a vector subspace of W. Solution: We must show that (a) 0W ∈ T(V1 ), (b) w1 + w2 ∈ T(V1 ), for all w1 , w2 ∈ T(V1 ), and (c) cw1 ∈ T(V1 ...
Facts about finite fields
... the ring K can be represented as the set of all polynomials with degree less than d and coefficients in F . Addition is carried out as usual for polynomials. When multiplying two polynomials, the relation f (x) = 0 is used to reduce terms of degree d or greater. As an additive group, K is isomorphic ...
... the ring K can be represented as the set of all polynomials with degree less than d and coefficients in F . Addition is carried out as usual for polynomials. When multiplying two polynomials, the relation f (x) = 0 is used to reduce terms of degree d or greater. As an additive group, K is isomorphic ...
3.4 Isomorphisms - NIU Math Department
... that φ is a well-defined function, and then prove that φ is a group isomorphism. Solution: If ([n]30 , [m]2 ) and ([k]30 , [j]2 ) are equal elements of Z30 ×Z2 , then 30 | n−k and 2 | m−j. It follows that 10 | n−k, and so [n]10 = [k]10 . Furthermore, 30 | 4(n−k), so 6 | 4(n−k), and then 6 | 3(m−j), ...
... that φ is a well-defined function, and then prove that φ is a group isomorphism. Solution: If ([n]30 , [m]2 ) and ([k]30 , [j]2 ) are equal elements of Z30 ×Z2 , then 30 | n−k and 2 | m−j. It follows that 10 | n−k, and so [n]10 = [k]10 . Furthermore, 30 | 4(n−k), so 6 | 4(n−k), and then 6 | 3(m−j), ...
Partial Sums of Powers of Prime Factors
... Lemma 3.2. If s ≥ 2 then there exist primes pi and integers ai , ni for i = 1, . . . , t satisfying the conditions of Lemma 3.1. Proof. Observe that t − 1 ≥ 3. Choose primes p1 , . . . , pt−1 such that pi ≡ 11 (mod 12) for all i = 1, . . . , t − 1, gcd(pi , pj − 1) = 1 for all i, j in {1, . . . , t ...
... Lemma 3.2. If s ≥ 2 then there exist primes pi and integers ai , ni for i = 1, . . . , t satisfying the conditions of Lemma 3.1. Proof. Observe that t − 1 ≥ 3. Choose primes p1 , . . . , pt−1 such that pi ≡ 11 (mod 12) for all i = 1, . . . , t − 1, gcd(pi , pj − 1) = 1 for all i, j in {1, . . . , t ...
4A. Definitions
... example, the element of G,,, pictured in 4.1 is order-preserving. If instead of the given 0 we had used a- b’, b-c’, cy a’, then the resulting element would be cyclicorder-preserving but not order-preserving. And if we had used a-b’, b-a’, c-c’, then the resulting element would be neither order-pres ...
... example, the element of G,,, pictured in 4.1 is order-preserving. If instead of the given 0 we had used a- b’, b-c’, cy a’, then the resulting element would be cyclicorder-preserving but not order-preserving. And if we had used a-b’, b-a’, c-c’, then the resulting element would be neither order-pres ...
1 Groups
... In the last result, the associativity of ∗ is definitely used in the proof. In fact the result is not in general true for nonassociative binary operations. ...
... In the last result, the associativity of ∗ is definitely used in the proof. In fact the result is not in general true for nonassociative binary operations. ...
Birkhoff's representation theorem
This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), and Birkhoff's HSP theorem representing algebras as products of irreducible algebras. Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices.