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.

Expert Solution

Colby Messinger answered 2 years ago

(1) Let G be a group and H, K be subgroups of G. (a) Show that...

(1) Let G be a group and H, K be subgroups of G. (a) Show that if H is a normal subgroup, then HK = {xy|x ? H, y ? K} is a subgroup of G. (b) Show that if H and K are both normal subgroups, then HK is also a normal subgroup. (c) Give an example of subgroups H and K such that HK is not a subgroup of G.

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

Let H and K be subgroups of a group G so that for all h in H and k in K there is a k' in K with hk = k'h. Proposition 2.3.2 shows that HK is a group. Show that K is a normal subgroup of HK.

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

Suppose G is a group and H and H are both subgroups of G. Let HK={hk, h∈H and k ∈K} a.give a example such that |HK| not equal to |H| |K| b. give a example to show f :HK →H ⨯K given by f(hk) = (h,k) may not be well defined.

Let G be a group and H and K be normal subgroups of G. Prove that...

Let G be a group and H and K be normal subgroups of G. Prove that H ∩ K is a normal subgroup of G.

The question is: Let G be a finite group, H, K be normal subgroups of G,...

The question is: Let G be a finite group, H, K be normal subgroups of G, and H∩K is also a normal subgroup of G. Using Homomorphism theorem ( or First Isomorphism theorem) prove that G/(H∩K) is isomorphism to a subgroup of (G/H)×(G/K). And give a example of group G with normal subgroups H and K such that G/(H∩K) ≆ (G/H)×(G/K), with explanation. I was trying to find some solutions for the isomorphism proof part, but they all seems to...

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 the cyclic group {[0], [1], [2], ..., [n − 1]} be denoted by Z/nZ. Consider...

Let the cyclic group {[0], [1], [2], ..., [n − 1]} be denoted by Z/nZ. Consider the following statement: for every positive integer n and every x in Z/nZ, there exists y ∈ Z/nZ such that xy = [1]. (a) Write the negation of this statement. (b) Is the original statement true or false? Justify your answer.

Question

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

Solutions

Expert Solution

Related Solutions