Order-cum-power statistics of a finite group
Let be a finite group. The order-cum-power statistics of give information about the number of elements of any given order that are powers of a certain kind. More precise formulations are given below.
In all the versions below, we let be the order of and be the set of natural numbers dividing .
Doubly cumulative version
In this version, the statistics are given by the following function : is the number of elements such that is the identity element and there exists such that .
If two finite groups have the same order-cum-power statistics, we say that they are order-cum-power statistics-equivalent finite groups.