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 well defined and construct the addition table for G/H with the operation ⊕.

Expert Solution

Colby Messinger answered 1 year ago

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

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

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