Relations admitting a transitive group of automorphisms
R. I. Tyshkevich
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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Mathematics of the USSR-Sbornik, 1975, 26:2, 245–259
MSC: 04A05, 05C25, 05C30, 20B25
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
\paper Relations admitting a~transitive group of automorphisms
\jour Mat. Sb. (N.S.)
\jour Math. USSR-Sb.
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