[SOLVED] Abstract algebra

Abstract algebra

Problem 1. Let n ∈ N and consider the set

i.e., Rn is the set of n-th roots of unity.

(a) Prove that R,, is a subgroup of (C”, x); i.e., the group of non-zero complex numbers under multiplication.

(b) Compute |Rnl.

(c) When n=8, prove that i ∈ Rg and compute lil.

(d) Prove that R is a cyclic group for every n ∈ N and name a generator for R, (a generator for R, is called a primitive n-th root of unity).

Problem 2. Let G be a group and let S be a non-empty, finite subset of G (I underlined “set” because weare not assuming that S is a subgroup). Further suppose that S satisfies the property

ab ∈ S whenever a ∈ S and b ∈ S; i.e., S is closed under the operation of G.

Prove that if z ∈ S, then z-1 ∈ S. Hint: consider the cyclic subgroup of G generated by x.