Let M and N be normal subgroups of G. Show that M∩N and MN are normal subgroups of G.
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Let M and N be normal subgroups of G. Show that M∩N and MN are normal subgroups of G.
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- Is W a subspace of the vector space? If not, state why. (Select all that apply.) w is the set of all vectors in R° whose components are Pythagorean triples. (Assume all components of a Pythagorean triple are positive integers.) O w is a subspace of R°. O w is not a subspace of R because it is not closed under addition. O w is not a subspace of R³ because it is not closed under scalar multiplication.Let G be a group and H be a subgroup of G.(a) Show that H is a normal subgroup of G if and only if Hx = xH for all x ∈ G.(b) Show that if |G : H| = 2, then H is a normal subgroup of G.4. (a) Let H be a subgroup of Sn with H not contained in An. Prove that [H: HnAn] =2. (b) Let G be a finite simple group with a subgroup H such that [G: H] = k > 2. Prove that G divides.
- Let E : y2 = x3 + 3x + 4 be an elliptic curve over F37. (a) Find all the elements of the elliptice curve group.(b) Find the order of the group.(c) Find a primitive element of this group and call it G.(d) Compute [30]G = G ⊕ G ⊕ · · · ⊕ G (addition of 30 many G’s) kindly help with full explanation. Thank you!Given the adjacency matrix of an undirected simple graph G = (V, E) mapped in a natural fashion onto a mesh of size n2, in Ο(n) time it can be decided whether or not G is bipartite prove it.Let (a1, b1) and (a2, b2) be open intervals (a1 < b1 and a2 < b2). Show that the Cartesian product (a1,b1) × (a2, b2) is an open region.
- 4. Consider the set V of vectors (x, X2, X3, X4) ER such that X1 + x3 = 0 and x, + x4 = 0. a) Prove that V is a subspace of R*. b) Give a basis and the dimension of V.(a) Draw an undirected graph G1 represented by the following adjacency matrix: 0 1 0 0 0 0 0 1 0 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 1 0 0 1 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 Does G1 contain a Hamiltonian cycle? Explain (b) Determine which pairs of graphs G1 in part (a) and G2 and G3 shown below are isomorphic and which are not, and explain your answer. If you use any invariant property in your argument, explain why this property is an invariant. Tex our answer. t 2 G2 G3 5. 7 8.Suppose that C is a linear code with Hamming weight 3 and that C'is one with Hamming weight 4. What can C' do that C can’t?
- Find a basis for and the dimension of the subspace w of R4. w = {(0, 4t, t, -t): t is a real number} (a) a basis for the subspace W of R4 (b) the dimension of the subspace W of R4Let C be a binary linear code. Show that either every member of Chas even weight or exactly half the members of C have even weight.Let g ZZ be defined by g(m) = m + 5. Is Is g a bijection? Explain. If g is a bijection, find its inverse.