Average order in wreath products

TL;DR

This paper derives an exact formula for the average order of elements in wreath products of finite groups, with specific focus on $p$-groups, and explores applications to limits of average orders in sequences of $p$-groups.

Contribution

It provides a new exact formula for average element order in wreath products and analyzes their distribution, especially for $p$-groups, advancing understanding of their algebraic properties.

Findings

Exact formula for average order in wreath products.

Estimate for average order of wreath products of $p$-groups.

Distribution of element orders in wreath products.

Abstract

We obtain an exact formula for the average order of elements of a wreath product of two finite groups. Then focussing our attention on $p$-groups for primes $p$, we give an estimate for the average order of a wreath product $A\wr B$ in terms of maximum order of elements of $A$ and average order of $B$ and an exact formula for the distribution of orders of elements of $A\wr B.$ Finally, we show how wreath products can be used to find several rational numbers which are limits of average orders of a sequence of $p$-groups with cardinalities going to infinity.