Prove that h is a normal subgroup of g and compute the

Question

Note: If G and G' are groups, G ⊕ G' is the direct product of G and G'.

1. Let G = D₁₂ ⊕ U(16) and H = ((r¹⁰; 5)), where r  ∈ D₁₂ is an element of order 12.

a) Prove that H is a normal subgroup of G.

b) Compute the order of the factor group G=H.

c) Compute the order of the element (r⁴; 3)H in the factor group G=H.

d) Prove or disprove that G=H is abelian.

e) Prove or disprove that G=H contains an element of order 8. (Note: This can be done without computing the orders of all elements of G=H. Think about relations between orders of elements of G and elements of G=H.)

2. Prove or disprove that D₁₂ is isomorphic to Z₃ ⊕ D₄.

Attachment:- MATH_Question.pdf