A corollary is that a permutation σ chosen uniformly at random
from the symmetric group S_{n} will almost surely not be an
automorphism of a Steiner triple system of order n, a quasigroup of
order n or a 1-factorisation of the complete graph K_{n}. Nor will
σ be one component of an autotopism for any Latin square of
order n.

For groups of order n it is known that automorphisms must have order less than~n, but we show that quasigroups of order n can have automorphisms of order greater than n. The smallest such quasigroup has order 7034. We also show that quasigroups of prime order can possess autotopisms that consist of three permutations with different cycle structures.

Our results answer three questions originally posed by D. Stones.