A short and simple derivation of the formula of Frobenius, which gives the dimensions of the irreducible representations of S n , the symmetric group on any number, n , of symbols, is given. These dimensions are the characters of the identity element of the group, i.e., of the element all of whose cycles are unary. It is shown how a slight modification of Frobenius' formula yields, when n = 2 p is even, the characters of an element of S n all of whose cycles are binary and, when n = 3 p is a multiple of 3, the characters of an element of S n all of whose cycles are ternary and, generally, when n = kp is a multiple of any positive integer k , the characters of an element of S n all of whose cycles are of length k . It is noteworthy that the calculations become simpler, rather than more complicated, as k increases. Finally, this paper shows how to derive from Frobenius' formula the characters of an element of S n which has at least one unary cycle and, from the present modifications of Frobenius' formula, the characters of an element of S n which has at least one cycle of length k, k = 2, 3,..., n .