## Mathematics, Department of

## Date of this Version

1993

## Citation

Published in * Communications in Algebra,* 21:9 (1993), pp 3259-3275.

doi 10.1080/00927879308824728

## Abstract

In 1990 Hendrik W. Lenstra, Jr. asked the following question: if *G* is a transitive permutation group of degree *n* and *A* is the set of elements of *G* that move every letter, then can one find a lower bound (in terms of n) for *f*(*G*) = |*A*|/|*G*|? Shortly thereafter, Arjeh Cohen showed that 1/*n* is such a bound.

Lenstra’s problem arose from his work on the number field sieve. A simple example of how *f*(*G*) arises in number theory is the following: if *h* is an irreducible polynomial over the integers, consider the proportion:

|{primes ≤ x | h has no zeroes mod p}| / |{primes ≤ x}|

As x → ∞, this ratio approaches *f*(*G*), where *G* is the Galois group of *h *considered as a permutation group on its roots.

Our results in this paper include explicit calculations of* f*(*G*) for groups *G* in several families. We also obtain results useful for computing *f*(*G*) when *G* is a wreath product or a direct product of permutation groups. Using this we show that {*f*(*G*) | *G* is transitive} is dense in [0, 1]. The corresponding conclusion is true if we restrict *G* to primitive groups.

## Comments

Copyright 1993 Marcel Dekker, Inc.; published by Taylor & Francis. Used by permission.