FOCAL SUBGROUPS OF FUSION SYSTEMS AND CHARACTERS OF HEIGHT ZERO

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1 FOCAL SUBGROUPS OF FUSION SYSTEMS AND CHARACTERS OF HEIGHT ZERO SEJONG PARK Abstract. For each finite p-group P, Glauberman [8] defined characteristic subgroup K (P ) and K (P ) of P and showed that, under suitable conditions, they control fusion and transfer in any finite group G having P as a Sylow p-subgroup. Recently Kessar and Linckelmann [11] showed that the control of fusion property of K (P ) and K (P ) generalizes to arbitrary fusion systems F on a finite p-group P, in particular to arbitrary blocks B having P as a defect group (the block case was proven earlier together with Robinson [12]). In this talk, we are going to (1) introduce the language of fusion systems, (2) show that the control of transfer property of K and K also generalizes to arbitrary fusion systems, and (3) discuss an application to the number of characters of height zero in a block by Robinson [23]. This is a joint work with Antonio Díaz, Adam Glesser, and Nadia Mazza [7]. 1. Fusion systems Fusion systems are categories satisfying certain axioms which mimic essential features of fusion in finite groups. Definition 1.1. Let P be a finite p-group. A fusion system F on P is a category whose objects are the subgroups of P and such that for each pair Q, R P, we have Hom P (Q, R) Hom F (Q, R) Inj(Q, R) (where composition of morphisms is the usual composition of maps), which satisfies the following properties: (1) For any ϕ Hom F (Q, R), the induced group isomorphism Q = ϕ(q) and its inverse are morphisms in F. (2) (Sylow axiom) Aut P (Q) is a Sylow p-subgroup of Aut F (Q) for any fully F-normalized subgroup Q of P. (3) (Extension axiom) Every morphism ϕ: Q P in F such that ϕ(q) is fully F- normalized extends to a morphism ψ : N ϕ P in F. Inj(Q, R) is the set of injective group homomorphisms from Q to R. Hom P (Q, R) is the set of group homomorphisms from Q to R induced by conjugations by some elements of P. R P is fully F-normalized if N P (R) N P (α(r)) for all α: R P in F. (Similarly, R P is fully F-centralized if C P (R) C P (α(r)) for all α: R P in F.) N ϕ = {y N P (Q) ϕ c y Q ϕ 1 Aut P (ϕ(q))}. Remark 1.2. Axioms of fusion systems were first formulated by Puig [21] in the early 1990s under the name of Frobenius category. The formulation and terminology given in Definition 1.1 (which is equivalent to Puig s) is due to Broto, Levi, and Oliver [5] (and its Date: August 31,

2 2 SEJONG PARK minimalization by Stancu [24]), except that our fusion system is what they call a saturated fusion system. A good introduction to the theory of fusion systems can be found in [18]. Example 1.3. Let G be a finite group with a Sylow p-subgroup P. Define F = F P (G) as the category whose objects are the subgroups of P and such that for each pair Q, R P, we have Hom F (Q, R) = Hom G (Q, R). Then we can show that F is a fusion system on P. The only nontrivial part is the extension axiom, which can be proven by a Frattini type argument. See [18, 2.11] for a proof. Example 1.4. Similarly, a p-block of a finite group induces a fusion system on its defect group. This construction uses Brauer pairs of a block, introduced by Alperin and Broué [1]. See [10, 3] for a detailed explanation. Remark 1.5. Not every fusion system F on a finite p-group P can be realized by a finite group(i.e. F = F P (G) for some finite group G having P as a Sylow p-subgroup). The most prominent example up to this point is the Solomon fusion system F Sol(q) on a Sylow 2-subgroup of Spin 7 (q) for q odd prime power (Levi, Oliver [15][16]; Kessar, Stancu [13]). On the other hand, every fusion system can be realized by an infinite group (Robinson [22]; Leary, Stancu [14]). As in finite groups, we can define local subsystems of a given fusion system. They make possible local analysis on fusion systems. Definition 1.6. Let F be a fusion system on a finite p-group P and let Q P. (1) N F (Q) is the category whose objects are the subgroups of N P (Q) and such that for each pair R 1, R 2 P, we have Hom NF (Q)(R 1, R 2 ) = { ψ R1 ψ Hom F (QR 1, QR 2 ), ψ(r 1 ) R 2, ψ(q) = Q }. (2) C F (Q) is the category whose objects are the subgroups of C P (Q) and such that for each pair R 1, R 2 P, we have Hom CF (Q)(R 1, R 2 ) = { ψ R1 ψ Hom F (QR 1, QR 2 ), ψ(r 1 ) R 2, ψ Q = id Q }. Proposition 1.7 (Puig [21, 2.14]; [18, 3]). Let F be a fusion system on a finite p-group P and let Q P. (1) If Q is fully F-normalized, then N F (Q) is a fusion system on N P (Q) (2) If Q is fully F-centralized, then C F (Q) is a fusion system on C P (Q) Example 1.8. Let F = F P (G) where G is a finite group with Sylow p-subgroup P and let Q P. (1) Q is fully F-normalized if and only if N P (Q) is a Sylow p-subgroup of N G (Q); in this case, N F (Q) = F NP (Q)(N G (Q)). (2) Q is fully F-centralized if and only if C P (Q) is a Sylow p-subgroup of C G (Q); in this case, C F (Q) = F CP (Q)(C G (Q)). Remark 1.9. One can further define when a fusion system is simple, and show that the Solomon fusion system F Sol(3) is a simple fusion system (Linckelmann [17]). Recently Aschbacher [3][2] set about a project towards the classification of simple 2-fusion systems, with a slightly weaker notion of simplicity than that of Linckelmann.

3 2. Focal subgroups of fusion systems In a finite group G, the focal subgroup of a Sylow p-subgroup P of G determines the existence of a nontrivial p-factor group of G. Since focal subgroups are determined solely by fusion in G on P, one can generalize the concept and theory of focal subgroups to fusion systems. Definition 2.1. Let G be a finite group with Sylow p-subgroup P. The group P G is called the focal subgroup of P with respect to G. Proposition 2.2 ([9, 3]). Let G be a finite group with Sylow p-subgroup P. (1) P G = xy 1 x, y P, y = gxg 1 for some g G. (2) P/P G = G/O p (G)G. (3) G has a normal subgroup of p-power index if and only if P G < P. Definition 2.3. Let F be a fusion system on a finite p-group P. The group is called the F-focal subgroup of P. [P, F] = ϕ(u)u 1 u P, ϕ Hom F ( u, P ). Theorem 2.4 (Broto, Levi, Oliver [4, 4.3]). A fusion system F on a finite p-group P has a fusion subsystem of p-power index if and only if [P, F] < P. Let P be a finite p-group. Glauberman [8, 12] defines characteristic subgroups K (P ) and K (P ) of P which satisfy certain complicated commutator relations, and shows that they satisfy a nice control of transfer property. Theorem 2.5 (Glauberman [8, 12.4]). Let p 5. Let P be a finite p-group and let K = K (P ) or K (P ). Then for every finite group G having P as a Sylow p-subgroup, we have P G = P N G (K). We generalize this property to arbitrary fusion systems. Its proof follows the usual line of taking a minimal counterexample and conducting local analysis on it. The proof also depends on a deep theorem of Broto, Levi, Oliver [5, 5.5] on the existence of a P -P -biset associated with a fusion system to produce the transfer map for fusion systems. Theorem 2.6 (Díaz, Glesser, Nadia, Park [7, 1.2]). Let p 5. Let P be a finite p-group and let K = K (P ) or K (P ). Then for every fusion system F on P, we have [P, F] = [P, N F (K)]. We finish this section with a corollary which finds an application in the next section. Corollary 2.7 (Díaz, Glesser, Nadia, Park [7, 6.1]). Let p 5. Let F be a fusion system on a finite p-group P 1 such that Aut F (P ) is a p-group. Then we have [P, F] < P. Proof. Suppose that the corollary is false and take a counterexample F with minimal number F of morphisms. By Theorem 2.6 and the minimality of F, we have F = N F (K). Let F = F/K be the category whose objects are subgroups of P = P/K and whose morphisms are induced from morphisms of F in the obvious way. It is well known that the category F is a fusion system on P ([18, 6.2]). If K < P, then P 1 and F < F, whence [P, F] = [P, F] < P ; it follows that [P, F] < P, a contradiction. If K = P, then by the assumption that Aut F (P ) is a p-group and by the fusion system version of Frobenius normal p-complement theorem [18, 1.4], we have F = N F (P ) = F P (P ) and hence [P, F] = [P, P ] < P, which is also a contradiction. 3

4 4 SEJONG PARK 3. An application: characters of height zero Throughout this section, let B be a p-block of a finite group G which has a maximal B- Brauer pair (P, e) with P 1, and let F = F (P,e) (G, B) be the fusion system on P induced by B through the Brauer pairs contained in (P, e). As usual, let Irr(B) denote the set of the ordinary irreducible characters of B, and let IBr(B) denote the set of irreducible Brauer characters of B. Also let Irr 0 (B) be the subset of Irr(B) consisting of characters of height zero. Let χ Irr(B). Let S be a set of representatives of G-conjugacy classes of Brauer elements (u, f) contained in (P, e). By Brauer s second main theorem, we have χ = χ (u,f) (u,f) S where χ (u,f) is a class function on G such that if v C G (u) is a p -element, then χ (u,f) (uv) = d u χϕϕ(v). ϕ IBr(f) Let λ Irr(P/[P, F]). We define a class function λ χ = λ(u)χ (u,f). (u,f) S This procedure is a special case of the Broué-Puig star contruction. Theorem 3.1 (Broué, Puig [6]). Let λ Irr(P/[P, F]), χ Irr(B). Then λ χ Irr(B). Robinson [23] observes that the star construction preserves character degrees (and hence heights), and also if λ Irr(P/[P, F]), for every χ Irr 0 (B) we have λ χ = χ if and only if λ = 1. Consequently, Theorem 3.2 (Robinson [23]). If [P, F] < P, then Irr 0 (B) 0 mod p. Now combining this with Corollary 2.7, we get the following result. Corollary 3.3 (Robinson [23, 4]). If Aut F (P ) is a p-group, then Irr 0 (B) 0 mod p. Proof. Let G p = p a and P = p d. Recall that we are assuming that d 1. A theorem of Brauer [19, 3.28] tells us that ( ) χ(1) 2 p a d 0 mod p d. In particular, we have χ Irr(B) χ Irr 0 (B) χ(1) 2 p 0 mod p. If p = 2, 3, then χ(1) 2 p = 1 for every χ Irr 0 (B) and hence Irr 0 (B) 0 mod p. (We don t need the assumption that Aut(P ) is a p-group here.) If p 5, then the desired result follows from Corollary 2.7 and Theorem 3.2. Note that this result is compatible with a special case of Alperin-McKay conjecture. Alperin-McKay Conjecture. Let B be a p-block of a finite group G with defect group P, and let b be the p-block of N G (P ) such that b G = B. Then we have Irr 0 (B) = Irr 0 (b).

5 Namely, in the situation of Corollary 3.3, we have Irr 0 (B) Irr 0 (b) mod p. To see this, let F be a fusion system on P induced by B. Observe that N F (P ) is the fusion system on P of b, and that the assumption that Aut F (P ) is a p-group means that N F (P ) = F P (P ). In other words, b is a nilpotent block in this case, and hence OGb is Morita equivalent to OP, where O is the usual complete discrete valuation ring of characteristic p (Puig [20]). Thus we have Irr 0 (b) 0 mod p. References 1. J. L. Alperin and Michel Broué, Local methods in block theory, Ann. of Math. (2) 110 (1979), no. 1, Michael Aschbacher, The generalized fitting subsystem of a fusion system, preprint. 3., Normal subsystems of fusion systems, preprint. 4. C. Broto, N. Castellana, J. Grodal, R. Levi, and B. Oliver, Extensions of p-local finite groups, Trans. Amer. Math. Soc. 359 (2007), no. 8, (electronic). 5. Carles Broto, Ran Levi, and Bob Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), no. 4, (electronic). 6. Michel Broué and Lluís Puig, Characters and local structure in G-algebras, J. Algebra 63 (1980), no. 2, Antonio Díaz, Adam Glesser, Nadia Mazza, and Sejong Park, Control of transfer and weak closure in fusion systems, preprint (2008). 8. G. Glauberman, Global and local properties of finite groups, Finite simple groups (Proc. Instructional Conf., Oxford, 1969), Academic Press, London, 1971, pp Daniel Gorenstein, Finite groups, second ed., Chelsea Publishing Co., New York, Radha Kessar, Introducton to block theory, Group representation theory, EPFL Press, Lausanne, 2007, pp Radha Kessar and Markus Linckelmann, ZJ-theorem for fusion systems, Trans. Amer. Math. Soc. 360 (2008), Radha Kessar, Markus Linckelmann, and Geoffrey R. Robinson, Local control in fusion systems of p-blocks of finite groups, J. Algebra 257 (2002), no. 2, Radha Kessar and Radu Stancu, A reduction theorem for fusion systems of blocks, J. Algebra 319 (2008), no. 2, Ian J. Leary and Radu Stancu, Realising fusion systems, Algebra Number Theory 1 (2007), no. 1, Ran Levi and Bob Oliver, Construction of 2-local finite groups of a type studied by Solomon and Benson, Geom. Topol. 6 (2002), (electronic). 16., Correction to: Construction of 2-local finite groups of a type studied by Solomon and Benson [Geom. Topol. 6 (2002), (electronic); mr ], Geom. Topol. 9 (2005), (electronic). 17. Markus Linckelmann, Simple fusion systems and the Solomon 2-local groups, J. Algebra 296 (2006), no. 2, , Introduction to fusion systems, Group representation theory, EPFL Press, Lausanne, 2007, pp G. Navarro, Characters and blocks of finite groups, London Mathematical Society Lecture Note Series, vol. 250, Cambridge University Press, Cambridge, Lluís Puig, Nilpotent blocks and their source algebras, Invent. Math. 93 (1988), no. 1, Lluis Puig, Frobenius categories, J. Algebra 303 (2006), no. 1, Geoffrey R. Robinson, Amalgams, blocks, weights, fusion systems and finite simple groups, J. Algebra 314 (2007), no. 2, , On the focal defect group of a block, characters of height zero, and lower defect group multiplicities, preprint (2008). 24. Radu Stancu, Equivalent definitions of fusion systems, preprint (2003). 5