In this paper we prove that there exists no function F (m, p) (where the first argument is an integer and the second a prime) such that, if G is a finite permutation p-group with m orbits, each of size at least pF (m,p), then G contains a fixed-point-free element. In particular, this gives an answer to a conjecture of Peter Cameron; see [4], [6].
Crestani, E., Spiga, P. (2010). Fixed-point-free elements in p-groups. ISRAEL JOURNAL OF MATHEMATICS, 180(1), 413-424 [10.1007/s11856-010-0109-7].
Fixed-point-free elements in p-groups
SPIGA, PABLO
2010
Abstract
In this paper we prove that there exists no function F (m, p) (where the first argument is an integer and the second a prime) such that, if G is a finite permutation p-group with m orbits, each of size at least pF (m,p), then G contains a fixed-point-free element. In particular, this gives an answer to a conjecture of Peter Cameron; see [4], [6].
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