SEMINAR: Groups and Combinatorics Seminar: Marvin Krings, 4pm Nov 23 in Robert Street LT

Speaker: Marvin Krings (RWTH, Aachen)

Title: The p-part of the order of an almost simple group of Lie type

Time and place: 4pm Friday 23 Nov 2018, Robert Street LT (not Weatherburn LT)

Abstract: Primitive permutation groups are fundamental building blocks in the sense that every finite permutation group can be built from the primitive ones. Apart from the alternating group A_n and the symmetric group S_n of degree n, the primitive subgroups G of S_n are small. For example, in 1980 Praeger and Saxl showed that |G| e 4^n, which is much smaller than n!/2. Since this time, powerful results such as the O’Nan-Scott Theorem, which classifies the primitive permutation groups, and the Classification of the Finite Simple Groups, have become available. We will bound the p-part |G|_p of |G| for some prime p. This is, the largest p-power p^{ u_p(G)} that divides |G|. The bound |G| e 4^n implies nu_p(G) e n og_p(4). We prove the stronger bound nu_p(G) e frac{2sqrt{n}}{(p-1)}+1 (with five exceptions). For several cases, we even obtain a bound that is logarithmic in n. Our proof uses the O'Nan-Scott theorem to reduce to simple groups. The hardest case, and the one I will discuss, is when the simple group is of Lie type.