COUNTING CONJUGACY CLASSES OF CYCLIC SUBGROUPS FOR FUSION SYSTEMS

SEJONG PARK

Abstract. 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≤_{G}H},

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≤_{G}S}

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: July 2013.

Theorem 3. Let F be a saturated fusion system on a finite p-group S. Let Ω be a characteristic biset for F. Then the rank of the matrix

(|P \Ω/Q|)_{P,Q≤_{F} S}

of the number of (P, Q)-orbits of Ω indexed by the F-conjugacy classes of subgroups of S is equal to the number of the F-conjugacy classes of cyclic subgroups of S.

Finally, one can replace the characteristic biset Ω in the above theorem by the characteristic idempotent ω_{F} (which is a virtual (S, S)-biset; see Section 3) with |P \ω_{F}/Q| as the linearized number of (P, Q)-orbits.

Theorem 4. Let F be a saturated fusion system on a finite p-group S. Let ω_{F} be the characteristic idempotent for F. Then the rank of the matrix

(|P \ω_{F}/Q|)_{P,Q≤_{F} S},

is equal to the number of the F-conjugacy classes of cyclic subgroups of S.

We will give a proof of Theorem 1 (and hence obtain Theorem 2 as a corollary), which is slightly different from that of [4]. This new proof uses (at least explicitly) only the Burnside ring B(G) of G, not the rational representation ring R_{ℚ}(G) as in [4]. Therefore it is better suited for adapting to the fusion system case (Theorem 3 and 4), which we do subsequently.

2. The group case

We prove Theorem 1. As remarked in Section 1, Theorem 2 then immediately follows as a corollary.

Let G be a finite group. Let B(G) be the Burnside ring of G, i.e., the Grothendieck ring of the isomorphism classes [X] of finite G-sets X. As an additive group, B(G) is a free abelian group with the canonical basis {[G/P] | P ≤_{G} G}. Let ℚB(G) = ℚ ⊗_{ℤ} B(G) and regard B(G) as a subgring of ℚB(G). In particular the canonical basis for B(G) is a ℚ-basis for ℚB(G).

It is a well-known fact that for each P ≤ G the fixed-point map

χ_{P} : B(G) → ℤ, [X] ↦ |X^{P} |,

is a ring homomorphism which depends only on the G-conjugacy class of P, and their product (tensored with ℚ)

χ = ℚ ⊗_{ℤ} ∏_{P ≤_{G}G} χ_{P} : ℚB(G) → ∏_{P ≤_{G}G} ℚ

is a ℚ-algebra isomorphism. For each P ≤ G, let e_{P}^{G} denote the element of ℚB(G) such that

χ_{Q}(e_{P}^{G}) = \begin{cases}1, & P =_{G} Q, \\ 0, & otherwise. \end{cases}

Then again the element e_{P}^{G} depends only on the G-conjugacy class of P and {e_{P}^{G} | P ≤_{G} G} is a set of pairwise orthogonal primitive idempotents of ℚB(G) whose sum is equal to 1; in particular it is a ℚ-basis for ℚB(G). Furthermore, for H ≤ G, let B(G)_{H} be the subgroup of B(G) generated by the elements [G/P] with P ≤_{G} H. Then ℚB(G)_{H} = ℚ ⊗_{ℤ} B(G)_{H} is a subalgebra of ℚB(G) with ℚ-basis {[G/P] | P ≤_{G} H}. Note that the elements e_{P}^{G} with P ≤_{G} H belong to ℚB(G)_{H} and hence {e_{P}^{G} | P ≤_{G} H} is another basis for ℚB(G)_{H}.

For each P ≤ G consider the ℚ-linear map

ρ_{P} : ℚB(G) → ℚ, [X] ↦ |P \X|,

which counts the P -orbits. By Burnside’s orbit counting lemma, we have

ρ_{P} (x) = \frac{1}{|P |} ∑_{u∈P} χ_{⟨u⟩}(x), x ∈ ℚB(G).

Thus

(1)  ρ_{P} (e_{Q}^{G}) ≠ 0 ⟺ Q is cyclic and Q ≤_{G} P .

Now the given matrix in Theorem 1 is equal to

(ρ_{P} (G/Q))_{P,Q≤_{G}H}.

By change of basis, this matrix has the same rank as

(ρ_{P} (e_{Q}^{G}))_{P,Q≤_{G}H}.

List the subgroups of H (up to G-conjugacy) in two blocks, the first consisting of cyclic subgroups and the second of noncyclic subgroups, and with nondecreasing order in each block. Then by (1) the above matrix has the form

(\begin{matrix}A & 0 \\ B & 0 \end{matrix})

where A is a lower triangular matrix with nonzero diagonal entries. Thus Theorem 1 follows.

3. The fusion system case

We first prove Theorem 3. In fact, we prove a slightly generalized version of it.

Proposition 5. Let F be a saturated fusion system on a finite p-group S. Suppose that Ω is a finite (S, S)-biset which is F-stable and F-generated and which contains the obvious (S, S)-biset S. Then the rank of the matrix

(|P \Ω/Q|)_{P,Q≤_{F} S}

is equal to the number of the F-conjugacy classes of cyclic subgroups of S.

We first explain the terminology. Let F be a saturated fusion system on a finite p-group S. An S-set X is F-stable if, for all P ≤ S and all F-morphism φ: P → S, the restrictions of the S-action on X to P via the inclusion P ↪ S and via φ: P → S give isomorphic P -sets. We say that an (S, S)-biset is F-stable if it is F × F-stable viewed as a left S × S-set by inverting the right action of S. An (S, S)-biset is F-generated if, viewed as a left S × S-set, all its isotropy subgroups are of the form Δ(P, φ) = {(u, φ(u)) | u ∈ P } with P ≤ S, φ: P → S in F. A finite (S, S)-biset Ω is called a characteristic biset for F if it is F-stable and F-generated and such that |Ω|/|S| is not divisible by p. It is easy to see that every characteristic biset Ω contains the (S, S)-biset S.

Define

B(F) = {x ∈ B(S) | χ_{P} (x) = χ_{P′}(x) for all P, P′ ≤ S with P =_{F} P′}.

Clearly B(F) is a subring of B(S), which is called the Burnside ring of the fusion system F. For a finite S-set X, we have [X] ∈ B(F) if and only if X is F-stable. As before let ℚB(F) = ℚ ⊗_{ℤ} B(F). Clearly the elements

e_{P}^{F} := ∑_{P′=_{F} P} e_{P′}^{S},

where P ≤ S and the sum is over the S-conjugacy classes of subgroups P′ of S which are F-conjugate to P, belong to ℚB(F). The set {e_{P}^{F} | P ≤_{F} S} is a set of pairwise orthogonal primitive idempotents of ℚB(F) whose sum is equal to 1; in particular it is a ℚ-basis for ℚB(F). By (1), we have

(2)  ρ_{P} (e_{Q}^{F}) ≠ 0 ⟺ Q is cyclic and Q ≤_{F} P .

Let Ω be the (S, S)-biset given in the above proposition. By the F-stability of Ω, the left S-set Ω/P of the right P -orbits of Ω is also F-stable for P ≤ S. Moreover

χ_{Q}([Ω/P]) ≠ 0 ⟹ Q ≤_{F} P; χ_{P} ([Ω/P]) ≥ |N_{S}(P)/P |.

The former follows from that Ω is F-generated and the latter from that Ω contains S. Hence

{[Ω/P] | P ≤_{F} S}

is a ℚ-basis for ℚB(F). Thus the matrix

(|P \Ω/Q|)_{P,Q≤_{F} S} = (ρ_{P} ([Ω/Q]))_{P,Q≤_{F} S}

has the same rank as

(ρ_{P} (e_{Q}^{F}))_{P,Q≤_{F} S},

which is equal to the number of the F-conjugacy classes of cyclic subgroups of S by (2).

Remark. Note that the finite group G in Theorem 2, viewed as an (S, S)-biset, satisfies the hypotheses for Ω in Proposition 5. Thus Theorem 2 can also be obtained from Proposition 5.

Now we address Theorem 4. In Proposition 5, the condition that Ω contains the (S, S)-biset S is equivalent to that χ_{P} (Ω/P) ≠ 0 for all P ≤ S, given the other conditions on Ω. Proposition 5 then applies to all virtual (S, S)-bisets ω with coefficients in ℚ which are F-stable, F-generated and such that χ_{P} (ω/P) ≠ 0 for all P ≤ S, where ω/P denotes the linearized right P -orbits of ω. The proof is identical to the one given above. In particular, Reeh [3, Proposition 4.5, Corollary 5.8] shows that if ω is the characteristic idempotent of F, i.e., the unique virtual (S, S)-biset with coefficients in ℤ_{(p)} which is F-stable, F-generated and which is an idempotent in the double Burnside ring ℤ_{(p)}B(S, S), then the elements ω/P = ω ∘_{S} [S/P] = β_{P} with P ≤_{F} S form a basis of ℤ_{(p)}B(F) such that χ_{P} (ω/P) ≠ 0. This proves Theorem 4.

Acknowledgements. We thank Prof. Markus Linckelmann for pointing us to the paper of Thévenaz and Prof. Thévenaz for helpful discussions.

References

[1] C. Broto, R. Levi and B. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), no. 4, 779–856 (electronic).

[2] S. Park, Realizing a fusion system by a single finite group, Arch. Math. (Basel) 94 (2010), no. 5, 405-410.

[3] S. Reeh, Transfer and characteristic idempotents for saturated fusion systems, preprint (2013).

[4] J. Thévenaz, A characterization of cyclic groups, Arch. Math. (Basel) 52 (1989), no. 3, 209– 211.

Section de mathématiques, École Polythechnique Fédérale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland

   E-mail address: sejong.park@epfl.ch