The Order of Automorphisms of Quasigroups
The Order of Automorphisms of Quasigroups
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DOI：
10.1002/jcd.21389
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发表时间：
2015-07
影响因子：
0.7
通讯作者：
B. McKay;Ian M. Wanless;Xiande Zhang
中科院分区：
文献类型：
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作者：
B. McKay;Ian M. Wanless;Xiande Zhang
我们证明了拟群或拉丁方的任意自同构的阶上的二次上界，从而也证明了Steiner三重系统的任意自同构的阶上的二次上界或完全图的1‐分解的二次上界。一个推论是，从对称群Sn中均匀随机选择的排列σ几乎肯定不是n阶Steiner三重系统、n阶拟群或完全图Kn的1‐分解的自同构。对于n阶的拉丁方，σ也不会是自同构的一个分量。对于n阶的群，已知自同构的阶必须小于n，但我们证明了n阶的拟群可以具有大于n的自同构。最小的此类拟群的阶为7034。我们还证明了素阶的拟群可以具有由三个不同循环结构的排列组成的自自洽。我们的结果回答了D. Stones最初提出的三个问题。
We prove quadratic upper bounds on the order of any autotopism of a quasigroup or Latin square, and hence also on the order of any automorphism of a Steiner triple system or 1‐factorization of a complete graph. A corollary is that a permutation σ chosen uniformly at random from the symmetric group Sn will almost surely not be an automorphism of a Steiner triple system of order n, a quasigroup of order n or a 1‐factorization of the complete graph Kn . 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.