Normalizers of Primitive Permutation Groups

2016/03/01 by Guralnick, Robert M., Maróti, Attila, Pyber, László
#12F10 #20B15 #20B35 #20C20 #20C99 #20D06 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1603.00187

Abstract

Let G be a transitive normal subgroup of a permutation group A of finite degree n. The factor group A/G can be considered as a certain Galois group and one would like to bound its size. One of the results of the paper is that |A/G| < n if G is primitive unless n = 3⁴, 5⁴, 3⁸, 5⁸, or 3¹⁶. This bound is sharp when n is prime. In fact, when G is primitive, |Out(G)| < n unless G is a member of a given infinite sequence of primitive groups and n is different from the previously listed integers. Many other results of this flavor are established not only for permutation groups but also for linear groups and Galois groups.

Related

• Base sizes of primitive permutation groups
• Base sizes of primitive groups: bounds with explicit constants
• On the minimal degree and base size of finite primitive groups
• Permutation groups and normal subgroups
• Base sizes of primitive groups of diagonal type