Groups with many roots

Given a prime p, a finite group G and a non-identity element g, what is the largest number of /pth roots g can have? We write /myrop(G), or just /myrop, for the maximum value of 1|G||{x∈G:xp=g}|, where g ranges over the non-identity elements of G. This paper studies groups for which /myrop is large. If there is an element g of G with more /pth roots than the identity, then we show /myrop(G)≤/myrop(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 /myrop(G)≤1p, while if G is a p-group of maximal class, then /myrop(G)≤1p+1p2 (both these bounds are sharp). We classify the groups with high values of /myro2, and give partial results on groups with high values of /myro3. Keywords /pth roots square roots cube roots