Let G = Z4 ⊕ Z4, and H = {(0, 0), (2, 0), (0, 2), (2,...

Let G = Z4 ⊕ Z4, and H = {(0, 0), (2, 0), (0, 2), (2, 2)}, and K = (1, 2). Is G/H isomorphic to Z4 or Z2 ⊕ Z2? Is G/K isomorphic to Z4 or Z2 ⊕ Z2?

Expert Solution

Colby Messinger answered 2 years ago

Let G = Z4 × Z4, H = ⟨([2]4, [3]4)⟩. (a) Find a,b,c,d∈G so that G...

Let G = Z4 × Z4, H = ⟨([2]4, [3]4)⟩. (a) Find a,b,c,d∈G so that G is the disjoint union of the 4 cosets a+H,b+ H, c + H, d + H. List the elements of each coset. (b) Is G/H cyclic?

Consider G = (Z12, +). Let H = {0, 3, 6, 9}. a. Show that H...

Consider G = (Z12, +). Let H = {0, 3, 6, 9}. a. Show that H is a subgroup of G. b. Find all the cosets of H in G and denote this set by G/H. [Note: If x ∈ G then H +12 [x]12 = {[h + x]12?? | [h]12 ∈ H} is the coset generated by x.] c. For H +12 [x]12, H +12 [y]12 ∈ G/H define (H+12[x]12)⊕(H+12[y]12) by(H+12 [x]12)⊕(H+12 [y]12)=H+12 [x+y]12. d. Show that ⊕ is...

Let G = Z4XZ3XZ2 and consider the two cyclic subgroups H = h(0; 1; 1)i and...

Let G = Z4XZ3XZ2 and consider the two cyclic subgroups H = h(0; 1; 1)i and K = h(2; 1; 1)i of G. (a) Find all cosets (along with the elements they contain) to H and K, respectively. (b) Write down Cayley tables for the factor groups G=H and G=K, and classify them according to the Fundamental Theorem of Finite Abelian Groups.

Let G, H, K be groups. Prove that if G ≅ H and H ≅ K...

Let G, H, K be groups. Prove that if G ≅ H and H ≅ K then G ≅ K.

Let G be a finite group and H a subgroup of G. Let a be an...

Let G be a finite group and H a subgroup of G. Let a be an element of G and aH = {ah : h is an element of H} be a left coset of H. If b is an element of G as well and the intersection of aH bH is non-empty then aH and bH contain the same number of elements in G. Thus conclude that the number of elements in H, o(H), divides the number of elements...

Question

Let G = Z4 ⊕ Z4, and H = {(0, 0), (2, 0), (0, 2), (2,...

Solutions

Expert Solution

Related Solutions

Let G = Z4 × Z4, H = ⟨([2]4, [3]4)⟩. (a) Find a,b,c,d∈G so that G...

Consider G = (Z12, +). Let H = {0, 3, 6, 9}. a. Show that H...

Let G = Z4XZ3XZ2 and consider the two cyclic subgroups H = h(0; 1; 1)i and...

Let G, H, K be groups. Prove that if G ≅ H and H ≅ K...

Let G be a finite group and H a subgroup of G. Let a be an...

Let G, H be groups and define the relation ∼= where G ∼= H if there...

Let G be a group and K ⊂ G be a normal subgroup. Let H ⊂...

Suppose G is a group and H and H are both subgroups of G. Let HK={hk,...

Let LaTeX: GG be an abelian group. Let LaTeX: H = { g \in G \mid...

Let H and K be subgroups of a group G so that for all h in...