Permutations

We begin with a consideration of multiply transitive actions. Numbers will be derived that allow directly to see from the cycle structure of the group elements if the action is multiply transitive or not. Afterwards we shall enumerate permutations with prescribed algebraic and combinatorial properties. We consider roots in finite groups, which means that we take a fixed natural number k and ask for the number of group elements x, the k-th power of which is equal to a given element g of the group G, xk = g. The case when g = 1 is of particular interest. Then we restrict attention to the symmetric group, in order to derive expressions for the number of roots in terms of characters and to show how permutrizations can be applied. It will be shown that the function which maps a permutation onto the number of its k-th roots is in fact a proper character of the symmetric group in question.