# Order-cum-power statistics of a finite group

## Definition

Let $G$ be a finite group. The order-cum-power statistics of $G$ 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 $n$ be the order of $G$ and $D$ be the set of natural numbers dividing $n$.

### Doubly cumulative version

In this version, the statistics are given by the following function $f: D \times D \to \mathbb{N}$: $f(d_1,d_2)$ is the number of elements $g \in G$ such that $g^{d_1}$ is the identity element and there exists $h \in G$ such that $h^{d_2} = g$.

## Related notions

If two finite groups have the same order-cum-power statistics, we say that they are order-cum-power statistics-equivalent finite groups.