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**1 - 2**of**2**### ALGORITHMIC RECOGNITION OF ACTIONS OF 2-HOMOGENEOUS GROUPS ON PAIRS

, 1998

"... We give an algorithm that takes as input a transitive permutation group (G, Ω) of degree n = �m � 2, and decides whether or not Ω is Gisomorphic to the action of G on the set of unordered pairs of some set Ɣ on which G acts 2-homogeneously. The algorithm is constructive: if a suitable action exists, ..."

Abstract
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We give an algorithm that takes as input a transitive permutation group (G, Ω) of degree n = �m � 2, and decides whether or not Ω is Gisomorphic to the action of G on the set of unordered pairs of some set Ɣ on which G acts 2-homogeneously. The algorithm is constructive: if a suitable action exists, then one such will be found, together with a suitable isomorphism. We give a deterministic O(snlogc n) implemention of the algorithm that assumes advance knowledge of the suborbits of (G, Ω). This leads to deterministic O(sn²) and Monte-Carlo O(snlogc n) implementations that do not make this assumption.