groups with many roots

Given a prime p, a finite group g and a non-identity element g, what is the largest number of p th roots g can have? we write ϱp(g), or just ϱp, for the maximum value of 1/g| |{x ∈ g : x^p = g}|, where g ranges over the non-identity elements of g. this paper studies groups for which ϱp is large. if there is an element g of g with more p th roots than the identity, then we show ϱp(g) ≤ ϱp(p), where p is any sylow p-subgroup of g, meaning that we can often reduce to the case where g is a p-group. we show that if g is a regular p-group, then ϱp(g) ≤ 1 p , while if g is a p-group of maximal class, then ϱp(g) ≤ 1/p + 1/p^2 (both these bounds are sharp). we classify the groups with high values of ϱ2, and give partial results on groups with high values of ϱ3.