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Course 111: Algebra, 22nd November 2006 To be handed in at tutorials on Nov 27th and 28th. 1. Consider the set of elements G = {0, 1, 2, 3, 4, 5} which forms a cyclic group under addition modulo 6. List the proper subgroups of G and the generators of the group. (Recall that {G} and {e}, where e = 0 here, are improper subgroups) Determine the left and right cosets of H = {0, 3} in G and construct the Cayley table of the quotient group G|H. 2. 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.