On the orders of primitive groups with restricted nonabelian composition factors

Bounds on finite quasiprimitive permutation groups

“…(A section of a group G is a quotient group of a subgroup of G.) The first result of this nature was due to Babai, Cameron and Palfy [4] in 1982. They obtained a polynomial bound on the order of primitive groups which satisfied certain restrictions on their non-cyclic composition factors.…”

“…It was shown in [28,Theorem A] that this action of alternating and symmetric groups on pairs was essentially the only obstruction for a linear bound for numbers of cycles less than p. The result in [28] is the best result of this type in the literature which does not depend on the finite simple group classification, and it holds with a small modification for quasiprimitive groups. 246 Cheryl E. Praeger and Aner Shalev [4] (ii) G = A n orS n ;…”

Maximal subgroups in finite and profinite groups

“…A typical example is the class S of all finite soluble groups, or the class F d (d 5) of all finite groups not involving the alternating group Alt d as a section; we call such groups Alt d -free. In general, if F is as above, then the groups in F satisfy the well-known Babai-Cameron-Pálfy restrictions on their nonabelian composition factors [3]. We say that a subgroup G Sym n is a maximal transitive (or primitive) F-subgroup if G is transitive (primitive), G ∈ F, and G is maximal with respect to these properties.…”

“…The reduction is based on the Babai-Cameron-Pálfy polynomial bound on the orders of Alt d -free primitive permutation groups [3]. Now, using the O'Nan-Scott Theorem [10] and [16, Theorem 1.1], we reduce Theorem 1.2 to the following result in finite linear groups.…”

“…Recall that there is a constant d such that all groups in F are Alt d -free. By [3] there is a constant c which depends only on d and such that |S| q c n . Now we can repeat the arguments of [7, p. 69].…”