Sejong Park

Tate's and Yoshida's theorems on control of transfer for fusion systems

We prove analogues of results of Tate and Yoshida on control of transfer for fusion systems. This requires the notions of

Analogues of Goldschmidt's thesis for fusion systems

p

Counting conjugacy classes of cyclic subgroups for fusion systems

-group residuals and transfer maps in cohomology for fusion systems. As a corollary we obtain a

INTRODUCTION TO FUSION SYSTEMS

p

The gluing problem for some block fusion systems

-nilpotency criterion due to Tate.

Glauberman's and Thompson's theorems for fusion systems

We extend the results of David Goldschmidts thesis concerning fusion in nite groups to saturated fusion systems. T 2 Syl 2 (G), then the exponent of Z(T ) (and hence of T ) is bounded by a function of the nilpotence class of T. He also includes in the write-up a fusion factorization result for an arbitrary nite group involving 0 1 Z and the Thompson subgroup. In this paper, we generalize these results to arbitrary saturated fusion systems. Throughout this paper, unless otherwise in- dicated, p denotes an arbitrary prime number, n a nonnegative integer, and P a nontrivial nite p-group. Theorem 1. Suppose P is of nilpotence class at most n(p 1) + 1 andF is a saturated fusion system on P with Op(F) = 1. Then Z(P ) has exponent at most p n .

Realizing a fusion system by a single finite group

Abstract. Thévenaz [Arch. Math. (Basel) 52 (1989), no. 3, 209–211] made an interesting observation that the number of conjugacy classes of cyclic subgroups in a finite group G is equal to the rank of the matrix of the numbers of double cosets in G. We give another proof of this fact and present a fusion system version of it. In particular we use finite groups realizing the fusion system ℱ as in our previous work [Arch. Math. (Basel) 94 (2010), no. 5, 405–410].