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Mathematics 1214: Introduction to Group Theory Homework exercise sheet 8 Due 12:50pm, Friday 26th March 2010 1. Let G be a group and let a, b ∈ G. (a) Prove that if a, b ∈ G, then a = b ⇐⇒ ab−1 = e. (b) Prove that G is an abelian group if and only if aba−1 b−1 = e for all a, b ∈ G. 2. [Optional question] Let G be a group and let a ∈ G. Prove that for any integers n, m ∈ Z, we have an am = an+m . [Suggestion: fix m ∈ Z and show that this works for n = 0 and n = 1, then prove it by induction on n for n ≥ 1. Finally, think about what happens when n < 0.] 3. Let G be a group and let a ∈ G. Prove that for any integers n, m ∈ Z, we have (an )m = anm . [Suggestion: fix n ∈ Z and show that this works for m = 0, and then prove it by induction on m for m > 0. Then consider what happens when m < 0.] 4. Disprove the following statements. (a) If G is a group and a, b ∈ G and n ∈ Z, then (ab)n = an bn . (b) If G is a group and a, b ∈ G and n ∈ Z, then (ab)n = bn an . 5. Let G be a group and let a ∈ G. (a) Show that hai = ha−1 i. (b) Deduce from (a) that o(a) = o(a−1 ). 6. For each element a in the group Z10 , compute o(a) and the cyclic subgroup hai. n 7. Let Q = { m : n, m ∈ Z, m 6= 0} denote the set of rational numbers. It is not hard to show that (Q, +) is an abelian group. Prove that (Q, +) is not a cyclic group. 8. Let n, m ∈ Z. [As usual, Z denotes the group (Z, +).] (a) Compute the cyclic subgroups hni and hmi, and show that hni ⊆ hmi ⇐⇒ m n. (b) Find a sequence of H1 , H2 , H3 , . . . of cyclic subgroups of Z such that H1 ) H2 ) H3 ) . . . .