Counting Conjugacy Classes of Cyclic Subgroups for Fusion Systems

We give another proof of an observation of Thévenaz [4] and present a fusion system version of it. Namely, for a saturated fusion system F on a finite p-group S, we show that the number of the F-conjugacy classes of cyclic subgroups of S is equal to the rank of certain square matrices of numbers of orbits, coming from characteristic bisets, the characteristic idempotent and finite groups realizing the fusion system F as in our previous work [2]. 1. Statements of the results In [4], Thévenaz observed the ‘curiosity’ that a finite cyclic group G can be characterized by the nonsingularity of the matrix of the numbers of double cosets in G. In fact He proved a more general fact that for an arbitrary finite group G the number of the conjugacy classes of cyclic subgroups of G is equal to the rank of that matrix. This can be stated slightly more generally by introducing a subgroup H of G and considering the G-conjugacy classes of subgroups of H as follows. Theorem 1. Let G be a finite group and let H ≤ G. The rank of the matrix (|P\G/Q|)P,Q≤GH , whose rows and columns are indexed by the G-conjugacy classes of subgroups of H and whose entries are the numbers of the corresponding double cosets in G, is equal to the number of the G-conjugacy classes of cyclic subgroups of H. In [2], we observed that every saturated fusion system F on a finite p-group S can be realized by a finite group G containing S as a (not necessarily Sylow) p-subgroup. Thus the above theorem yields a fusion system version as follows. Theorem 2. Let F be a saturated fusion system on a finite p-group S. Let G be a finite group which contains S as a subgroup and realizes F . Then the rank of the matrix (|P\G/Q|)P,Q≤GS is equal to the number of the F-conjugacy classes of cyclic subgroups of S. By a result of Broto, Levi and Oliver [1, Proposition 5.5], every saturated fusion system F on a finite p-group S has a (non-unique) characteristic biset Ω. See Section 3 for a precise definition; in particular, Ω is a finite (S, S)-biset, i.e., a finite set with compatible left and right S-actions. If F is the fusion system of a finite group G on its Sylow p-subgroup S, then G is a characteristic biset for F with the obvious S-action on the left and right. So we may well expect that the matrix of the above theorem with G replaced by Ω has the same rank. Indeed this is the case. Date: September 2, 2014. 1