1. Let : G→ G' be a homomorphism. (i) Show that if g E G has order n <∞ then the order of (g) divides n. [2 marks] (ii)

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1 Let G G Be A Homomorphism I Show That If G E G Has Order N Then The Order Of G Divides N 2 Marks Ii 1 (99.29 KiB) Viewed 27 times

1. Let : G→ G' be a homomorphism. (i) Show that if g E G has order n <∞ then the order of (g) divides n. [2 marks] (ii) We say (R, +,0) is the group of additive real numbers. What are the orders of its elements? List all the homomorphisms S3 → R. [3 marks] (iii) What are the orders of the elements of the multiplicative group of the reals (RX,, 1) ? List all the homomorphisms S3 → RX. [3 marks] (iv) Are the groups (R, +,0) and (RX, ., 1) isomorphic? Are the groups (R, +,0) and (R>0,, 1) isomorphic? Here (R> 1) is the multiplicative group of positive real numbers. [3 marks] (v) What is the order of the permutation (12345) (6 789) in S₁0? Can you find an element of order 12 in S7? Can you find an element of order 12 in S6? [3 marks]