Butterfly, Möbius, and Double Burnside algebras of noncyclic finite groups

The double Burnside ring B(G,G) of a finite group G is the Grothendieck ring of finite (G,G)-bisets with respect to the tensor product of bisets over G. Many invariants of the group G, such as the (single) Burnside ring B(G) and the character ring R_C(G), are modules over B(G,G). The double Burnside ring B(G,G) and its various subrings appear as crucial ingredients in functorial representation theory, homotopy theory, and the theory of fusion systems. It is known to be semisimple over rationals if and only if G is cyclic, and in this case an explicit isomorphism onto a direct product of full matrix algebras is given by Boltje and Danz (2013). But not much is know beyond that on the explicit algebra structure. We generalize some techniques of Boltje and Danz for cyclic groups to arbitrary finite groups and as a result obtain an explicit isomorphism of the rational double Burnside algebra of a finite group G into a block triangular matrix algebra when all Sylow subgroups (for all primes) of G are cyclic. Such groups can be characterized as groups G where the Zassenhaus Butterfly lemma gives the meet of two sections (H, K) of G with respect to the subsection relation. Key ingredients are a refinement of the inclusion relation among subgroups of G x G and Möbius inversion over various posets of subgroups. This is a joint work with Goetz Pfeiffer (Galway).Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Counting conjugacy classes of cyclic subgroups for fusion systems

We give another proof of an observation of Th\'evenaz \cite{T1989} and present a fusion system version of it. Namely, for a saturated fusion system \CF on a finite $p$-group $S$, we show that the number of the \CF-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 \CF as in our previous work \cite{P2010}.Comment: 5 page

Minimal characteristic bisets and finite groups realizing Ruiz-Viruel exotic fusion systems

Continuing our previous work, we determine a minimal left characteristic biset $X$ for every exotic fusion system $\mathcal{F}$ on the extraspecial group $S$ of order $7^3$ and exponent $7$ discovered by Ruiz and Viruel, and analyze the finite group $G$ obtained from $X$ which realizes $\mathcal{F}$ as the full subcategory of the $7$-fusion system of $G$. In particular, we obtain an upper bound for the exoticity index for $\mathcal{F}$.Comment: 22 page

Realizing fusion systems inside finite groups

We show that every (not necessarily saturated) fusion system can be realized as a full subcategory of the fusion system of a finite group. This result extends our previous work \cite{Park2010} and complements the related result \cite{LearyStancu2007} by Leary and Stancu.Comment: 3 page

Mackey functors and sharpness for fusion systems

We develop the fundamentals of Mackey functors in the setup of fusion systems including an acyclicity condition as well as a parametrization and an explicit description of simple Mackey functors. Using this machinery we extend Dwyer's sharpness results to exotic fusion systems $\mathcal{F}$ on a finite $p$-group $S$ with an abelian subgroup of index $p$.Comment: to appear in Homology, Homotopy, and Applications (HHA

Counting conjugacy classes of cyclic subgroups for fusion systems

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

Tate's and Yoshida's theorem 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 $p$-group residuals and transfer maps in cohomology for fusion systems. As a corollary we obtain a $p$-nilpotency criterion due to Tate.Comment: 20 page

On the cohomology of pro-fusion systems

We prove the Cartan-Eilenberg stable elements theorem and construct a Lyndon-Hochschild-Serre type spectral sequence for pro-fusion systems. As an application, we determine the continuous mod-$p$ cohomology ring of $\text{GL}_2(\mathbb{Z}_p)$ for any odd prime $p$.Comment: 19 pages, 4 figure