A Characteristic Subgroup for Fusion Systems Silvia Onofrei The Ohio State University in collaboration with Radu Stancu Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 1/24 Theorem [Kessar, Linckelmann, 2008] Let p be an odd prime and let W be a Glauberman functor. Let F be a fusion system on a finite p-group S. If F is a Qd(p)-free then F = NF (W (S)). ZJ-Theorem Let p be an odd prime. Let G be a Qd(p)-free finite group with CG(Op(G)) ≤ Op(G). Then Z(J(S)) is a normal subgroup of G. Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 2/24 ZJ-Theorem Let p be an odd prime. Let G be a Qd(p)-free finite group with CG(Op(G)) ≤ Op(G). Then Z(J(S)) is a normal subgroup of G. Theorem [Kessar, Linckelmann, 2008] Let p be an odd prime and let W be a Glauberman functor. Let F be a fusion system on a finite p-group S. If F is a Qd(p)-free then F = NF (W (S)). Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 2/24 Theorem A (O-Stancu, 2008) Let S be a finite 2-group. Let F be a Σ4-free fusion system over S. Then there exists a nontrivial characteristic subgroup W (S) of S with the property that F = NF (W (S)) Theorem [Stellmacher, 1990] Let S be a finite nontrivial 2-group. Then there exists a nontrivial characteristic subgroup W (S) of S which is normal in G, for every finite Σ4-free group G with S a Sylow 2-subgroup and CG(O2(G)) ≤ O2(G). Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 3/24 Theorem [Stellmacher, 1990] Let S be a finite nontrivial 2-group. Then there exists a nontrivial characteristic subgroup W (S) of S which is normal in G, for every finite Σ4-free group G with S a Sylow 2-subgroup and CG(O2(G)) ≤ O2(G). Theorem A (O-Stancu, 2008) Let S be a finite 2-group. Let F be a Σ4-free fusion system over S. Then there exists a nontrivial characteristic subgroup W (S) of S with the property that F = NF (W (S)) Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 3/24 Main Theorem (O-Stancu, 2008) Let p be a prime and let S be a finite p-group. There exists a Glauberman functor S −! 1 6= W (S) char S with the additional property that for every Qd(p)-free fusion system F on S F = NF (W (S)) Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 4/24 Outline of the Talk 1 Background on Fusion Systems 2 H-Free Fusion Systems 3 Characteristic Functors 4 The Characteristic Subgroup W(S) 5 Proof of Theorem A Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 5/24 P and Q are F -conjugate if Q ' j(P) for j 2 HomF (P;Q). The subgroup Q of S is: 0 fully F -centralized if jCS(Q)j ≥ jCS(Q )j for all Q0 ≤ S which are F -conjugate to Q. 0 fully F -normalized if jNS(Q)j ≥ jNS(Q )j for all Q0 ≤ S which are F -conjugate to Q. Background on Fusion Systems Let S be a finite p-group and let P;Q be subgroups of S. A Category F on S: objects are the subgroups of S; HomF (P;Q) ⊆ Inj (P;Q); any j 2 HomF (P;Q) is the composite of an F -isomorphism followed by an inclusion. Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 6/24 The subgroup Q of S is: 0 fully F -centralized if jCS(Q)j ≥ jCS(Q )j for all Q0 ≤ S which are F -conjugate to Q. 0 fully F -normalized if jNS(Q)j ≥ jNS(Q )j for all Q0 ≤ S which are F -conjugate to Q. Background on Fusion Systems Let S be a finite p-group and let P;Q be subgroups of S. A Category F on S: objects are the subgroups of S; HomF (P;Q) ⊆ Inj (P;Q); any j 2 HomF (P;Q) is the composite of an F -isomorphism followed by an inclusion. P and Q are F -conjugate if Q ' j(P) for j 2 HomF (P;Q). Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 6/24 Background on Fusion Systems Let S be a finite p-group and let P;Q be subgroups of S. A Category F on S: objects are the subgroups of S; HomF (P;Q) ⊆ Inj (P;Q); any j 2 HomF (P;Q) is the composite of an F -isomorphism followed by an inclusion. P and Q are F -conjugate if Q ' j(P) for j 2 HomF (P;Q). The subgroup Q of S is: 0 fully F -centralized if jCS(Q)j ≥ jCS(Q )j for all Q0 ≤ S which are F -conjugate to Q. 0 fully F -normalized if jNS(Q)j ≥ jNS(Q )j for all Q0 ≤ S which are F -conjugate to Q. Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 6/24 For a morphism j 2 HomF (Q;S) x y Nj = fx 2 NS(Q)j9y 2 NS(j(Q)); j( u) = j(u); 8u 2 Qg a subgroup with the property that: QCS(Q) ≤ Nj ≤ NS(Q). Background on Fusion Systems A fusion system on S is a category F on S such that: HomS(P;Q) ⊆ HomF (P;Q) for all P;Q ≤ S : AutS(S) is a Sylow p-subgroup of AutF (S). Every j : Q ! S such that j(Q) is fully F -normalized extends to a morphism jb : Nj ! S. Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 7/24 Background on Fusion Systems A fusion system on S is a category F on S such that: HomS(P;Q) ⊆ HomF (P;Q) for all P;Q ≤ S : AutS(S) is a Sylow p-subgroup of AutF (S). Every j : Q ! S such that j(Q) is fully F -normalized extends to a morphism jb : Nj ! S. For a morphism j 2 HomF (Q;S) x y Nj = fx 2 NS(Q)j9y 2 NS(j(Q)); j( u) = j(u); 8u 2 Qg a subgroup with the property that: QCS(Q) ≤ Nj ≤ NS(Q). Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 7/24 Background on Fusion Systems Lemma 1 Let F be a fusion system on S and let Q be a subgroup of S. a) There is j 2 HomF (NS(Q);S) such that j(Q) is fully F -normalized. b) If Q is fully F -normalized, then j(Q) is fully normalized, for any j 2 HomF (NS(Q);S). Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 8/24 Background on Fusion Systems The normalizer of Q in F is the category NF (Q): objects: the subgroups of NS(Q); morphisms: j 2 HomF (R;T ), for which there exists jb 2 HomF (QR;QT ) such that jbjQ 2 AutF (Q) and jbjR = j. • Q fully F -normalized =) NF (Q) a fusion system on NS(Q) The centralizer of Q in F is the category CF (Q): objects: the subgroups of CS(Q); morphisms: j 2 HomF (R;T ), for which there exists jb 2 HomF (QR;QT ) such that jbjQ = idQ and jbjR = j. • Q fully F -centralized =) CF (Q) a fusion systems on CS(Q) Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 9/24 Background on Fusion Systems The category NS(Q)CF (Q): objects: the subgroups of NS(Q); morphisms: group homomorphisms j : R ! T , for which there exists y : QR ! QT and x 2 NS(Q) such that yjQ = cx and yjR = j. • Q fully F -centralized =) NS(Q)CF (Q) a fusion systems on CS(Q) Q E F - if F = NF (Q). Op(F ) - the largest normal subgroup in F . If Q E F , F =Q is the fusion system on S=Q: morphisms: for Q ≤ P;R ≤ S, a group homomorphism y : P=Q ! R=Q is a morphism in F =Q if there is j 2 HomF (P;R) satisfying y(xQ) = j(x)Q for all x 2 P. Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 10/24 If Q is fully F -normalized, F -centric F -radical then: N (Q) = (LF ) F FNS(Q) Q Background on Fusion Systems F -centric if CS(j(Q)) ⊆ j(Q), for all j 2 HomF (Q;S). F -radical if Op(AutF (Q)) = Inn(Q). F is constrained if Op(F ) is F -centric. Theorem [BCGLO, 2005] If F is constrained then there exists a, unique up to isomorphism, 0 finite p -reduced p-constrained group L such that F = FS(L) and with S a Sylow p-subgroup of L. Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 11/24 Background on Fusion Systems F -centric if CS(j(Q)) ⊆ j(Q), for all j 2 HomF (Q;S). F -radical if Op(AutF (Q)) = Inn(Q). F is constrained if Op(F ) is F -centric. Theorem [BCGLO, 2005] If F is constrained then there exists a, unique up to isomorphism, 0 finite p -reduced p-constrained group L such that F = FS(L) and with S a Sylow p-subgroup of L.