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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
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