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 Mat. Sb. (N.S.), 1975, Volume 97(139), Number 2(6), Pages 262–277 (Mi msb3651)

Relations admitting a transitive group of automorphisms

R. I. Tyshkevich

Abstract: The concepts of a Cayley relation of arbitrary arity and a quotient relation are defined. Cayley relations are characterized as those relations whose automorphism groups contain regular subgroups. The freedom of Cayley relations is proved: any relation with a transitive automorphism group is isomorphic to a quotient relation of a Cayley relation.
Using Cayley relations, two problems are solved: 1) for a given transitive permutation group on a set $V$ to construct all relations on $V$ whose automorphism groups contain it; 2) for a given abstract group $G$ to construct all relations whose automorphism groups contain a transitive subgroup isomorphic to $G$.
Cayley relations are used to describe the graphs, digraphs, and tournaments having a transitive automorphism group. A solution is given for a weak variant of a problem of König: what is the nature of a transitive permutation group $G$ if there exists a nontrivial graph whose automorphism group contains $G$?
Finally, Cayley relations are used to describe the centralizer of a transitive permutation group in the symmetric group.
Bibliography: 23 titles.

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English version:
Mathematics of the USSR-Sbornik, 1975, 26:2, 245–259

Bibliographic databases:

UDC: 519.41
MSC: 04A05, 05C25, 05C30, 20B25

Citation: R. I. Tyshkevich, “Relations admitting a transitive group of automorphisms”, Mat. Sb. (N.S.), 97(139):2(6) (1975), 262–277; Math. USSR-Sb., 26:2 (1975), 245–259

Citation in format AMSBIB
\Bibitem{Tys75} \by R.~I.~Tyshkevich \paper Relations admitting a~transitive group of automorphisms \jour Mat. Sb. (N.S.) \yr 1975 \vol 97(139) \issue 2(6) \pages 262--277 \mathnet{http://mi.mathnet.ru/msb3651} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=398887} \zmath{https://zbmath.org/?q=an:0319.20006} \transl \jour Math. USSR-Sb. \yr 1975 \vol 26 \issue 2 \pages 245--259 \crossref{https://doi.org/10.1070/SM1975v026n02ABEH002479}