M.Sc. MACS we ask you to find the Sylow ?subgroups of the dihedral group ?ћ =⟨??∶ ?ћ,?,?ѥ?ѥ⟩

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In this exercise, we ask you to find the Sylow ?subgroups of the dihedral group ?ћ =⟨??∶ ?ћ,?,?ѥ?ѥ⟩,?∈ ℕ,?≥ 2. (a) Let ?be an odd prime that divides ? ?= ?џ? ?∤ ? Suppose ?= ⟨?љ⟩. Show that ?is the unique Sylow ?subgroup of ?ћ. (b) Prove the relation ?і?ї?ј?љ = { ?і?ї+?if k is even ?і+?ѥ?Ȓ?if k is odd. Further, find all the elements of order 2 in ?ћ. (c) Find all the Sylow 2-subgroups of ?ћ when ?is odd. Describe them in terms of ?and ? (d) Suppose ?is even, ?= 2?њ, where 2 ∤ ? ?≥ 2. Let ?= ⟨?њ⟩ and ?= ⟨?ߩ. Show that ?сisasubgroup of ?ћ. What is its order? (e) Suppose ?is as in the previous part. Find all the Sylow 2-supgroups of ?ћ. Describe them in terms of ?and ? (2) (4) (1) (3) (2) (3) (4) (2) (3) (3) #MSCMACS
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