Following Babai’s algorithm (Graph isomorphism in quasipolynomial time, arXiv:1512.03547v2, 2016) for the string isomorphism problem, we determine that it is possible to write expressions of short length describing certain permutation cosets, including all permutation subgroups. This is feasible both in the original version of the algorithm and in its CFSG-free version, by Babai (2016, §13.1) and Pyber (A CFSG-free analysis of Babai’s quasipolynomial GI algorithm, arXiv:1605.08266, 2016). The existence of such descriptions gives a weak form of the Cameron–Maróti classification, even without assuming CFSG. This is applicable to proofs of diameter bounds for \(\mathrm {Alt}(n)\) as in Helfgott (Growth in linear algebraic groups and permutation groups: towards a unified perspective, arXiv:1804.03049, 2018): our main result is used in Dona (Towards a CFSG-free diameter bound for \({\mathrm{Alt}}(n)\), arXiv:1810.02710v3, 2018) to free Helfgott’s proof from the use of CFSG. We also thoroughly explicate Babai’s recursion process (as given in Helfgott et al. in Graph isomorphisms in quasi-polynomial time, arXiv:1710.04574, 2017) and obtain explicit constants for the runtime of the algorithm, both with and without the use of CFSG.

遵循针对字符串同构问题的Babai算法（准多项式时间内的图形同构，arXiv：1512.03547v2,2016），我们确定可以编写描述某些置换陪集（包括所有置换子组）的较短长度的表达式。这在Babai（2016，§13.1）和Pyber（Babai拟多项式GI算法的无CFSG分析，arXiv：1605.08266，2016）的原始算法版本和无CFSG版本中都是可行的。即使没有假设CFSG，此类描述的存在也给出了Cameron-Maróti分类的较弱形式。这适用于\（\ mathrm {Alt}（n）\）的直径范围的证明如Helfgott（线性代数组和置换组中的增长：朝着统一的观点看，arXiv：1804.03049，2018年）：我们的主要结果用于Dona（朝着\（{\ mathrm {Alt}}的无CFSG直径约束（n）\），arXiv：1810.02710v3，2018年）从使用CFSG中释放Helfgott的证明。我们还彻底说明了Babai的递归过程（如Helfgott等人在准多项式时间内的Graph同构，arXiv：1710.04574,2017中给出），并获得了使用和不使用CFSG时算法运行时的显式常量。