Tag - Finite groups
Let G be a primitive permutation group on a finite set X and recall that a subset of X is a base for G if its pointwise stabiliser is trivial. The base size of G, denoted b(G), is defined to be the minimal size of a base. This natural invariant has been intensively studied for many years, finding a wide range of applications. In this talk I will report on recent progress concerning a project initiated by Jan Saxl in the 1990s, which seeks to determine all the primitive groups with b(G) = 2. I will also outline some of the main applications and I will highlight one or two related problems.
By a classical theorem of Jordan, every faithful transitive action of a nontrivial finite group admits a derangement (an element with no fixed points). More recently, the existence of derangements with additional properties has attracted much attention, especially for primitive actions of almost simple groups. Surprisingly, there exist almost simple groups with elements that are derangements in every faithful primitive action; we say that these elements are totally deranged. I'll talk about ongoing work to classify the totally deranged elements of almost simple groups, and I'll mention how this solves a question of Garzoni about invariable generating sets for simple groups.
I will discuss joint work with Bob Guralnick and Russ Woodroofe. We investigate invariable generation of finite simple groups by two elements of prime or prime power order. We apply our results to a problem raised by Ken Brown on the topology of the order complex of the poset of all cosets of all proper subgroups of an arbitrary finite group.
In this talk, I will introduce the notion of functorial equivalence of blocks of finite groups, developed in recent joint work with Deniz Yilmaz. For a commutative ring R, and a field k of characteristic p>0, we introduce the category of diagonal p-permutation functors over R. To a pair (G,b) of a finite group G and a block idempotent b of kG, we associate a diagonal p-permutation functor FG,b, and we say that two such pairs (G,b) and (H,c) are functorially equivalent over R if the functors FG,b and FH,c are isomorphic. We show that the category of diagonal p-permutation functors over an algebraically closed field of characteristic 0 is semisimple. We obtain a precise description of the simple functors, and explicit formulas for their multiplicities as summands of FG,b. It follows that functorial equivalence preserves the defect groups of blocks, their number of simple modules, and their number of ordinary irreducible characters. This also leads to characterizations of nilpotent blocks, and to a finiteness theorem in the spirit of Donovan's finiteness conjecture.
The Picard group Tk(G) of the stable module category of a finite group has been an important object of study in modular representation theory, starting with work Dade in the 1970s. Its elements are equivalence classes of so-called endotrivial modules, i.e., modules M such that End(M) is isomorphic to a trivial module direct sum a projective kG-module. 1-dimensional characters, and their shifts, are examples of such modules, but often exotic elements exist as well. My talk will be a guided tour of how to calculate Tk(G), using methods from homotopy theory. The tour will visit joint work with Tobias Barthel and Joshua Hunt, with Jon Carlson, Nadia Mazza and Dan Nakano, and with Achim Krause.
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