SEMINAR: Groups and Combinatorics Seminar: Saul Freeman (UWA) Maximally symmetric p-groups

Speaker: Saul Freeman (UWA)

Time and place: 16:00 Friday 28/04/2017 in ENCM: G04 Engineering LT2 (N.B. alternative venue!)

Title: Maximally symmetric p-groups

Abstract: The set of finite p-groups of nilpotency class r, rank d and exponent p includes a universal group P such that every group in this set is a quotient of P. For certain values of r and p, Bamberg, Glasby, Morgan and Niemeyer (BGMN) constructed P as a group whose underlying set is a Cartesian product of vector spaces, and proved some results about the structure of Aut(P). The main goal of BGMN's work was to determine the maximal subgroups H of the general linear group GL(d,p) for which there exists an associated p-group G with nilpotency class at most 4, rank d and exponent p, such that Aut(G) induces H on the Frattini quotient of G. BGMN showed that such a group G exists when p > 3, and when H lies in an Aschbacher class other than C_6 or C_9.

In this talk, I will present new results about the structure of Aut(P) for certain universal groups P. I will also give new examples of matrix subgroups H, which lie in class C_6 or C_9 and are not necessarily maximal, with associated p-groups G. In particular, I will show that for each odd prime p, there exists a p-group related to the exceptional group of Lie type G_2(p).

Past and future seminars may be found at http://www.maths.uwa.edu.au/~glasby/GandCSeminars.html