In this paper we will show how to generate A_(kn+1) and S_(kn+1) using a copy of S_n and an element of order k in A_(kn+1) and S_(kn+1) respectively, for all positive integers n ≥ 2 and all positive integers k ≥ 2. We will also show how to generate A_(kn+1) and S_(kn+1) symmetrically using n elements each of order k, for all n ≥ 2 and all even integers k ≥ 2.