Minimal normal subgroups of transitive permutation groups of square-free degree

• Published in: Discrete mathematics Vol. 307; no. 3; pp. 373 - 385
• Main Authors: Dobson, Edward, Malnič, Aleksander, Marušič, Dragan, Nowitz, Lewis A.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 06.02.2007
• Subjects: 2-Closed group; Semiregular automorphism; Square-free degree; Transitive permutation group; Vertex-transitive graph; Square-free degree; Vertex-transitive graph; Transitive permutation group; Semiregular automorphism; 2-Closed group
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2005.09.029

It is shown that a minimal normal subgroup of a transitive permutation group of square-free degree in its induced action is simple and quasiprimitive, with three exceptions related to A 5 , A 7 , and PSL ( 2 , 29 ) . Moreover, it is shown that a minimal normal subgroup of a 2-closed permutation group of square-free degree in its induced action is simple. As an almost immediate consequence, it follows that a 2-closed transitive permutation group of square-free degree contains a semiregular element of prime order, thus giving a partial affirmative answer to the conjecture that all 2-closed transitive permutation groups contain such an element (see [D. Marušič, On vertex symmetric digraphs, Discrete Math. 36 (1981) 69–81; P.J. Cameron (Ed.), Problems from the fifteenth British combinatorial conference, Discrete Math. 167/168 (1997) 605–615]).