Abstract
Let G be a permutation group on a finite set Ω. The k-closure G(k) of G is the largest subgroup of the symmetric group Sym(Ω) having the same orbits with G on the kth Cartesian power Ωk of Ω. The group G is called \({\textstyle{3 \over 2}}\)-transitive, if G is transitive and the orbits of a point stabilizer Gα on Ω{α} are of the same size greater than 1. We prove that the 2-closure G(2) of a \({\textstyle{3 \over 2}}\)-transitive permutation group G can be found in polynomial time in size of Ω. Moreover, if the group G is not 2-transitive, then for every positive integer k its k-closure can be found within the same time. Applying the result, we prove the existence of a polynomial-time algorithm for solving the isomorphism problem for schurian \({\textstyle{3 \over 2}}\)-homogeneous coherent configurations, that is coherent configurations naturally associated with \({\textstyle{3 \over 2}}\)-transitive groups.
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The authors were supported by the Russian Foundation for Basic Research (Grant 18-01-00752).
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Vasil’ev, A.V., Churikov, D.V. The 2-Closure of a \({\textstyle{3 \over 2}}\)-Transitive Group in Polynomial Time. Sib Math J 60, 279–290 (2019). https://doi.org/10.1134/S0037446619020083
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DOI: https://doi.org/10.1134/S0037446619020083