The following conditions on a group Gare equivalent (i) G is abelian (ii) (ab) 2 a 2 b 2 for all a,b G (iii) (ab) 1 a 1 b 1 for all a, b G (iv) (ab) n a n b n for all n c Z and all a,b G (v) (ab) n a n b n for three consecutive integers n and all a,b G Show that (v) (i) is false if three is replace...

To show that v i is false if three is replaced by two we need to find a group ...

The following conditions on a group Gare equivalent: (i) G is abelian; (ii) (ab) 2 = a

Question:

The following conditions on a group Gare equivalent: (i) G is abelian; (ii) (ab)² = a²b² for all a,b ϵ G; (iii) (ab)^-1 = a^-1b^-1 for all a, b ϵ G; (iv) (ab)ⁿ = aⁿbⁿ for all n c Z and all a,b ϵ G; (v) (ab)ⁿ = aⁿbⁿ for three consecutive integers n and all a,b ϵ G. Show that (v) ⇒ (i) is false if "three" is replaced by "two."