Math 322: Problems for Mastery

2.1. Definitions: groups, subgroups, homomorphisms. (1) Which of the following are groups? If yes, prove the group axioms. If not, show that an axiom fails. (a) The “half integers” 12Z = { a 2 | a ∈ Z } ⊂ Q, under addition. (b) The “dyadic integers” Z[ 12 ] = { a 2k | a ∈ Z, k ≥ 0 } ⊂ Q, under addition. (c) The non-zero dyadic integers, under multiplication. (2) [DF1.1.9] Let F = { a+ b √ 2 | a, b ∈ Q } ⊂ R. (a) Show that (F,+) is a group. (b) Show that (F \ {0} , ·) is a group. RMK: Together with the distributive law, (a),(b) make F a field. (3) Let G be a commutative group and let k ∈ Z. (a) Show that the map x 7→ x is a group homomorphism G→ G. (b) Show that the subsets G[k] = { g ∈ G | g = e } and { g | g ∈ G } are subgroups. RMK For a general group G let G = 〈{ g | g ∈ G }〉 be the subgroup generated by the kth powers. You have shown that, for a commutative group, G = { g | g ∈ G } .