Average order in regular wreath products

We obtain an exact formula for the average order of elements of regular wreath product of two finite groups. Then focussing our attention on p-groups for primes p, we give an estimate for the verage 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.

Keywords:

average order , wreath product , maximum order , order distribution , semidirect product