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Course 111: Algebra, 17th November 2006 1. Suppose G is a cyclic group of order 4, G = {e, a, a2 , a3 } and a4 = e. Find the order of each element of G. Find all the subgroups of G. List the distinct left cosets of the subgroup H = {e, a2 } in G. 2. Prove that for H and K both subgroups of a group G. The product, HK is itself a subgroup of G iff HK = KH. Note: HK = {x ∈ G|x = hk, h ∈ H, k ∈ K} 3. Recall the properties of cosets listed in your notes. From this list prove the following: For a subgroup H of a group G, • x ∈ xH for all x ∈ G. • if x and y ∈ G and if y = xa for some a ∈ H then xH = yH. • each left coset of H in G has the same number of elements as H.