Primitive permutation groups with a regular subgroup

Abstract

This paper starts the classification of the primitive permutation groups (G, ) such that G contains a regular subgroup X. We determine all the triples (G, , X) with soc(G) an alternating, or a sporadic or an exceptional group of Lie type. Further, we construct all the examples (G, , X) with G a classical group which are known to us. Our particular interest is in the 8-dimensional orthogonal groups of Witt index 4. We determine all the triples (G, , X) with soc(G) ∼ = P + 8 (q). In order to obtain all these triples, we also study the almost simple groups G with G ∼ = P 2n+1 (q). The case G ∼ = U n (q) is started in this paper and finished in [B. Baumeister, Primitive permutation groups of unitary type with a regular subgroup, Bull. Belg. Math. Soc. 112 (5) (2006) 657-673]. A group X is called a Burnside-group (or short a B-group) if each primitive permutation group which contains a regular subgroup isomorphic to X is necessarily 2-transitive. In the end of the paper we discuss B-groups.

Point regular groups of automorphisms of generalised quadrangles

“…In particular, we construct more regular subgroups than those described in Theorem 1.2; see Remark 3.17. The generalised quadrangle of order (2,4) The reader may notice that the existence of a regular group of automorphisms implies that the point graph is a Cayley graph with the same automorphism group as the generalised quadrangle. Moreover, since E is normal in the full automorphism group of the Cayley graph, they are normal Cayley graphs for E. However P is not normal in the full automorphism group of the Cayley graph, and so when q is not a power of 2 or 3, the point graph is a normal and non-normal Cayley graph for two isomorphic groups.…”

“…and equality holds since by (2), each element of Z (E) is a commutator. For p-groups, the Frattini subgroup is the smallest normal subgroup such that the quotient is elementary abelian, and so Φ(E) = E = Z (E).…”

“…Let Q be a finite thick classical generalised quadrangle and let G be a group of automorphisms that acts regularly on the points of Q. Then one of the following holds: (5,2) and G is an extraspecial group of order 27 and exponent 3. 2.…”

Regular subgroups of primitive permutation groups

“…The remaining cases (b, q) = (2, 8), (3,2), (4,2) are excluded in entirely the same fashion, and we leave this to the reader.…”

“…Li and Seress [28,Theorem 1.2] handle the special case where the degree |Ω| is square-free and the regular subgroup B lies in L. Regular subgroups of two sporadic groups (HS.2 and J 2 .2) have been found in [17,21]. Finally, some families of almost simple primitive groups have been dealt with independently by Baumeister in [4,5].…”