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Mathematics 1214: Introduction to Group Theory Tutorial exercise sheet 9 1. Let G1 and G2 be two groups. Prove that if H1 ≤ G1 and H2 ≤ G2 , then H1 × H2 ≤ G1 × G2 . [Here, H1 × H2 is the set {(h1 , h2 ) : h1 ∈ H1 , h2 ∈ H2 }.] 2. (a) Explain how the relationship Z ⊆ R allows us to identify Z × Z with a subset of the plane P = R2 = R × R, and use this to “draw a picture” of Z × Z. (b) Let G = Z × Z and let H = h(1, 2)i. Draw a picture of H, and draw a picture of the right coset Ha where a = (−2, 3). (c) The right cosets of H partition G. “Explain” this by drawing a picture. How many (different) right cosets of H are there? 3. If G and H are groups with |G| > 1 and |H| > 1, prove that there are at least four different subgroups of G × H. [Hint: consider subgroups of the form S × T where S ≤ G and T ≤ H]. 4. (a) Let G be a group. Prove that ∆ = {(g, g) : g ∈ G} is a subgroup of G × G. (b) Find a counterexample to the following statement: If G1 and G2 are groups, then H ≤ G1 × G2 =⇒ H = H1 × H2 for some subgroups H1 ≤ G1 and H2 ≤ G2 . [Hint: take G1 = G2 = Z2 and apply (a).] 5. Prove that if G is a finite group of prime order, then G is abelian. 6. A corollary to Lagrange’s theorem (Corollary 39) tells us that if n is a prime number, then every group of order n is cyclic. For each non-prime integer n with 4 ≤ n ≤ 14, give an example of a cyclic group of order n, and a non-cyclic group of order n. [Hint: the cyclic group should be easy. For the non-cyclic group, if n is even with n ≥ 6 then you should be able to think of a non-abelian group of order n, and then consider the remaining cases, n = 4 and n = 9 separately.] [Remark: the case n = 15 is different. Even though 15 is not prime, it turns out that every group of order 15 is cyclic!].