Find the order of the cyclic subgroup of C^x generated by 1+i. Solution The elements of this group are of the form: (1+i)^n This group has the same size as the order of 1+i in C^x. That\'s the smallest integer n>1 such that (1+i)^n = 1. However, there is no such n. That means the size of this subgroup is infinite. The reason is simple: (1+i) has magnitude ?(2). Every positive integer power of (1+i) will give you some number with magnitude ?2 ^n. That means the magnitude gets bigger, and can never come back down to 1. For more concrete evidence, you might notice that: (i+1)^(8n) = 2^(4n) which is clearly infinite, but those powers of 2 are all contained inside.