Groups problems

Problem 1.1. Prove that if G is an abelian group, then for all a, b ∈ G and all integers n, (a · b) n = a n · b n . Problem 1.2. If G is a group such that (a · b) 2 = a 2 · b 2 for all a, b ∈ G, show that G must be abelian. Problem 1.3. If G is a finite group, show that there exists a positive integer N such that a N = e for all a ∈ G. Problem 1.4. (1) If the group G has three elements, show it must be abelian. (2) Do part (1) if G has four elements. (3) Do part (2) if G has four elements Problem 1.5. Show that if every element of the group G is its own inverse, then G is abelian.

Subgroups &Cyclic groups

1.Show that Union of two subgroups need not be a subgroup 2..Show that (Z,+) is a cyclic Group 3. Write the generators of the group{1,w,w2} 4. Write the generators of the group{1,-1,i,-i} 5 Find the order of -1 and 3 in (Z,+).