Let D10 denote the dihedral group of the hexagon. Thus, D10 is generated by r and...

Let D₁₀ denote the dihedral group of the hexagon. Thus, D₁₀ is generated by r and f with r¹⁰=f²=1 and fr=r^-1f=r⁹f

(a) Show that D10 has a subgroups N and M such that
i. N ∼= D5 (isomorphic to D5)
ii. M is a cyclic subgroup group of order 2
iii. N ∩ M = {e}
iv. N and M are each normal in D10
v. Every element in g ∈ D10 is a product g = nm of elements n ∈ N and m ∈ M.

(b) Prove that if G is any group with normal subgroups N and M such that N ∩ M = {e}, then
nm = mn for all n ∈ N and m ∈ M.

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Expert Solution

Colby Messinger answered 1 year ago

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