Order-cum-power statistics of a finite group

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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.

Relation with other statistics

Weaker statistics