## Abstract

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 {ie11-1} 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*.

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## Additional information

Supported by ARC grants DP0662946 and DP1093320. Stones also partially supported by NSFC grant 61170301.

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### Cite this article

Browning, J., Stones, D.S. & Wanless, I.M. Bounds on the number of autotopisms and subsquares of a Latin square.
*Combinatorica* **33, **11–22 (2013). https://doi.org/10.1007/s00493-013-2809-1

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### Mathematics Subject Classification (2000)

- 05B15
- 20D45
- 20N05