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Mathematics 1214: Introduction to Group Theory Homework exercise sheet 9 College is closed on Friday 2nd April. Please go to my office, room 2.9 in the maths department some time on Thursday 1st April to hand in your work. 1. Let G and H be two groups. Prove that G × H is abelian if and only if G is abelian and H is abelian. [Hint: for “ =⇒ ”, first think about elements of the form (g, eH ).] 2. Let p, q be prime integers. Show that if G is a group with |G| = pq and H ≤ G, then either H = G, or H is a cyclic subgroup of G. [Remark: in mathematics, the statement “A or B” means that one of the three possibilities “A and not B”, “B and not A”, “A and B” is true. In other words, “A or B” can be thought of as “A, or B, or both”.] [Hint: if k ∈ N and p and q are primes with k < pq, then k|pq =⇒ k|p or k|q.] 3. Given the group G below, determine the number of (different) subgroups of G, and list them. Also determine whether or not G is a cyclic group. (a) G = Z2 × Z2 (b) G = Z2 × Z3 4. Let n ≥ 2 be a non-prime integer, and suppose that G is a cyclic group of order n with identity element e. Prove that there is a subgroup of G other than {e} and G. [This is a partial converse to Corollary 39]. 5. Let n ≥ 3 and write Dn = {ι, ρ, ρ2 , ρ3 , . . . , ρn−1 , r0 , r1 , r2 , . . . , rn−1 } where, as usual, ι is the identity element of Dn , rj is reflection in the line through the origin making an angle of π/n with the positive x-axis, and ρ is rotation by 2π/n counterclockwise. Consider the following subgroups of Dn : hιi, hr0 i, hr1 i, . . . , hrn−1 i, hρi, Dn . (a) Compute the order of each of these subgroups, and prove that no two of these subgroups are equal. (b) Prove that if n is prime then there are no other subgroups of Dn . [Hint: use Exercise 2] (c) Prove that if n is not prime, then there is a subgroup of Dn that is not in this list.