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A note on finite groups determined by a combinatorial property. (English) Zbl 1231.05298
Brualdi, Richard A. (ed.) et al., Combinatorics and graphs. Selected papers based on the presentations at the 20th anniversary conference of IPM on combinatorics, Tehran, Iran, May 15–21, 2009. Dedicated to Reza Khosrovshahi on the occasion of his 70th birthday. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4865-4/pbk). Contemporary Mathematics 531, 103-108 (2010).
Summary: One of important problems in the theory of finite groups is to decide whether two given groups are isomorphic. In this paper, some properties of the transitive permutation group $$\Gamma_G$$ whose orbitals coincide with the basis relations of the group association scheme $${\mathcal X}(G)$$ of a given finite group $$G$$ is investigated. Then it is proved that the group $$G$$ is determined uniquely (up to isomorphism) by the group association scheme of $$G$$, under the assumption that the permutation group $$\Gamma_G$$ is 2-closed.
For the entire collection see [Zbl 1202.05003].
##### MSC:
 05E30 Association schemes, strongly regular graphs 20B05 General theory for finite permutation groups
##### Keywords:
group association scheme; 2-closed