I have my set of elements in G (the fifth roots of unity) as {z^0, z^1, z^2, z^3, z^4} derived from de Moivre's formula. I am trying to prove G forms a group under complex multiplication. The only part I am stuck on is the Associative Property.
Since I have 5 elements, but 3 variables (a,b,c) for the associative property, I am trying to figure out how to set it up. The only example I have found is for the third roots of unity. Any help would be appreciated.
You need to show that for any $a,b,c\in G$, $(ab)c = a(bc)$.
To see this, note that when you multiply different powers of $z$, you add the exponents, and addition is associative.