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Bounds on the number of autotopisms and subsquares of a Latin square

A subsquare of a Latin square L is a submatrix that is also a Latin
square. An autotopism of L is a triplet of permutations
(α,β,γ) such that L is unchanged after the rows are
permuted by α, the columns are permuted by β and the
symbols are permuted by γ. Let n!(n-1)!R_{n} be the
number of n × n Latin squares. We show that an n × n
Latin square has at most n^{O(log k)} subsquares of order k
and admits at most n^{O(log n)} autotopisms. This enables us
to show that p^{n/(p-1)-O(log2 n)} divides
R_{n} for all primes p. We also extend a theorem by McKay and
Wanless that gave a factorial divisor of R_{n}, and give a new
proof that R_{p} = 1 mod p for prime p.