Finite p-groups that determine p-nilpotency locally Th. Weigel - PowerPoint PPT Presentation

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups Finite p-groups that determine p-nilpotency locally Th. Weigel UNIVERSITÀ DEGLI STUDI DI MILANO-BICOCCA Dipartimento di Matematica e Applicazioni Novosibirsk, July 19 th , 2013

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups p-Nilpotent finite groups or "boaring" finite groups . . . Definition Let p be a prime number. A finite group G is called p-nilpotent if for P ∈ Syl p ( G ) . G = P ⋉ O p ′ ( G ) Remark Let G be a finite p-nilpotent group. If H ⊆ G = ⇒ H is finite p-nilpotent. N ⊳ G = ⇒ G / N is finite p-nilpotent. If H is finite p-nilpotent = ⇒ G × H is finite p-nilpotent. A finite group Y is nilpotent if, and only if, Y is p-nilpotent for every prime p. For P ∈ Syl p ( G ) one has N G ( P ) = P × O p ′ ( N G ( P )) .

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups Cayley’s 2-nilpotency criterion A. Cayley (1821-1895) Theorem (A. Cayley) Let G be a finite group such that P ∈ Syl 2 ( G ) is cyclic. Then G is 2 -nilpotent. A. Cayley

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups Burnside’s p-nilpotency criterion W. Burnside (1852-1927) Theorem (W. Burnside (1911)) Let G be a finite group and let p be a prime number such that P ∈ Syl p ( G ) is abelian and N G ( P ) is p-nilpotent. Then G is p-nilpotent. W. Burnside

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups Thompson’s p-nilpotency criterion Definition For a finite p-group P one defines the Thompson subgroups by J R ( P ) = � A ⊆ P | A abelian, rk ( A ) maximal � , J 0 ( P ) = � B ⊆ P | B abelian, | B | maximal � . Theorem (J.G. Thompson (1964)) J.G. Thompson Let G be a finite group, let p be odd and let P ∈ Syl p ( G ) . Then t.f.a.e.: G is p-nilpotent; C G ( Z ( P )) and N G ( J R ( P )) are p-nilpotent.

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups Glauberman’s p-nilpotency criterion Theorem (G. Glauberman (1968)) Let G be a finite group, let p be odd and let P ∈ Syl p ( G ) . Then t.f.a.e.: G is p-nilpotent; N G ( Z ( J 0 ( P ))) is p-nilpotent. G. Glauberman

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups What can go wrong, will go wrong ..... (Murphy’s law) Wishful thinking Let G be a finite group, let p be a prime and let P ∈ Syl p ( G ) . ??? � N G ( P ) p-nilpotent G p-nilpotent . Example (a) Let V = V ( p , F p ) and Aff p ( F p ) = GL p ( F p ) ⋉ V . Put C p = Z / p Z . Let p be odd, and let T ◦ ⊂ GL p ( F p ) be a maximal split torus, i.e., | T | = ( p − 1 ) p . Then for G ◦ = ( C p ⋉ T ◦ ) ⋉ V one has for P ∈ Syl p ( G ) that N G ◦ ( P ) = P ≃ C p ≀ C p , but G ◦ is not p-nilpotent. Let T cox be a Coxeter torus, i.e., | T cox | = p p − 1. Then for G cox = ( C p ⋉ T cox ) ⋉ V and Q ∈ Syl p ( G cox ) one has again that N G cox ( Q ) = Q ≃ C p ≀ C p , but G cox is not p-nilpotent. (b) For G = GL n ( F 2 ) and P ∈ Syl 2 ( G ) one has that N G ( P ) = P . However, for n > 2 the group G is not 2-nilpotent.

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups Finite p-groups that determine p-nilpotency locally Definition (T.W.) A finite p-group P is said to determine p-nilpotency locally , if for every finite group G with Q ∈ Syl p ( G ) , Q ≃ P , N G ( Q ) p-nilpotent G p-nilpotent. = ⇒ Example (W. Burnside (1911)) Abelian p-groups determine p-nilpotency locally. Notation By DN p we denote the class of all finite p-groups that determine p-nilpotency locally.

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups p-central p-groups of height k Definition (A. Mann (1994)) For a finite p-group P let ζ m ( P ) , m ≥ 0, ζ 0 ( P ) = 1, denote the ascending central series of P , i.e., ζ m + 1 ( P ) /ζ m ( P ) = Z ( P /ζ m ( P )); and Ω 1 ( P ) = � g ∈ P | g p = 1 � . Then P is called p-central of height k , k ≥ 1, if Ω 1 ( P ) ⊆ ζ k ( P ) . Theorem (J. Gonzalez-Sanchez, T.W. (2011)) Let p be odd, and let P be p-central of height ≤ p − 1 . Then P determines p-nilpotency locally, i.e., P ∈ DN p .

� � � p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups Slim (and xslim) p-groups Definition (T.W.) For a prime number p let Y p ( 1 ) = C p ≀ C p . For m ≥ 2 let Y p ( m ) be the p-group which is isomorphic to the pull-back of the diagram � C p C p m β Y p ( m ) C p ≀ C p where β : C p ≀ C p → C p is the canonical map, and C p m = Z / p m Z . A p-group P is called slim , if ∀ U ⊆ P ∀ m ≥ 1 : U �≃ Y p ( m ) . A p-group P is called xslim , if ∀ U , V ⊆ P , V ⊳ U : U / V �≃ Y p ( 1 ) . By slim p (resp. xslim p ) we will denote the class of slim (resp. xslim) p-groups. In particular, xslim p � slim p

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups The Main Theorem Theorem (T.W. (2012)) For p odd one has slim p ⊆ DN p . For p = 2 one has xslim 2 ⊆ DN 2 . Theorem (T.W. (2012)) For p odd slim p ⊆ DN p is the maximal s-closed subclass of DN p . For p = 2 xslim 2 ⊆ DN 2 is the maximal s- and q-closed subclass of DN 2 .

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups Applications Applications Let P be a finite p-group. If cl ( P ) ≤ p − 1 , then P ∈ xslim p . If P is of exponent p, then P ∈ xslim p . . Hall) then P ∈ xslim p . If P is regular (in the sense of P If srk ( P ) ≤ p − 1 , where srk ( ) is the sectional rank, then P ∈ xslim p . If p is odd and P is p-central of height p-1, then P ∈ slim p .

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups The proof . . . or at least a sketch the usual suspect . . . and R. Brauer . . . pqp-sandwiches Let q be a prime coprime to p. Let Q be an irreducible (left) F q [ C p ] -module. Put G 0 = C p ⋉ Q. Let P 0 be an irreducible, non-trivial (left) F p [ G 0 ] -module. G = G 0 ⋉ P 0 will be called a pqp-sandwich group. Remark R. Brauer’s first Main Theorem implies that P 0 is a projective ¯ F p [ G 0 ] -module.

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups The proof, part II Schur-Frattini covers . . . Definition Let G be a finite group. An extension π : X → G is called a p-Frattini extension , if ker ( π ) ⊆ O p (Φ( G )) . A p-Frattini extension π : X → G will be called a p-Schur-Frattini extension if ker ( π ) ∩ O p ( X ) ⊆ Z ( O p ( X )) . Proposition Let p be odd, let G be a pqp-sandwich group and let π : X → G be a finite p-Schur-Frattini extension. Then there exists m ≥ 1 such that X contains a subgroup isomorphic to Y p ( m ) .

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups The proof, part III minimal counter examples . . . Proof of the Main Theorem.. sketchy, sketchy . . . If p = 2 then the minimal counterexample to the assertion is a 2q2-sandwich group. For p odd the minimal counterexample to the assertion is a p-Schur-Frattini cover of a pqp-sandwich group.

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups Tate’s p-nilpotency criterion J. Tate (1925*) Theorem (J. Tate (1964)) Let G be a finite group, let p be prime and let P ∈ Syl p ( G ) . Then G is p-nilpotent if, and only if, res G P ( ): H • ( G , F p ) − → H • ( P , F p ) is an isomorphism of rings. J. Tate Remark = ⇒ is ”just” the Hochschild-Lyndon-Serre spectral sequence.

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups Quillen’s p-nilpotency criterion D. Quillen (1940-2011) Theorem (D. Quillen (1971)) Let G be a finite group, let p be an odd prime and let P ∈ Syl p ( G ) . Then G is p-nilpotent if, and only if, res G P ( ): H • ( G , F p ) − → H • ( P , F p ) is an F-isomorphism of rings. D. Quillen Remark A homomorphism φ : A → B of finitely generated, graded commu- tative F p -algebras is called an F -isomorphism if ker ( φ ) consists of nilpotent elements and for all b ∈ B there exists n ∈ N such that b p n ∈ im ( φ ) .

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups Swan p-groups S.B. Priddy Definition (S.B. Priddy) A finite p-group P is called a Swan group , if for all finite groups G with P ≃ Q ∈ Syl p ( G ) the (injective) homomorphism of rings res G N G ( P ) ( ): H • ( G , F p ) − → H • ( Q , F p ) N G ( Q ) . is an isomorphism. Theorem (R.G. Swan) S.B. Priddy Finite abelian p-groups are Swan groups. Theorem (H-W. Henn, S.B. Priddy (1994)) Let p be odd. Then every finite p-central p-group is a Swan group.

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups Control of p-fusion the swiss connection ... G. Mislin, J. Thévenaz Definition Let G be a finite group, P ∈ Syl p ( G ) , and H ⊆ G such that P ⊆ H ⊆ G . If for all A , B ⊆ P and g ∈ G such that i g | A : A → B is an isomorphism there exists h ∈ H such that i h | A = i g | A : A → B , then H is said to control p -fusion in G . Theorem (G. Mislin, J. Thévenaz (1990, 1993)) Let P be a finite p-group. Then t.f.a.e.: P is a Swan group; N G ( Q ) controls p-fusion for every finite group G with P ≃ Q ∈ Syl p ( G ) .

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups Consequences following J. Thévenaz . . . Swan = ⇒ Burnside. (J. Thévenaz (1993)) p odd, P p-central and finite = ⇒ N G ( Q ) controls p -fusion for all finite groups G with P ≃ Q ∈ Syl p ( G ) . p odd: P p -central and finite = ⇒ P Swan. (J. Tate (1964)) p odd: P p -central and J. Thévenaz finite = ⇒ P ∈ DN p .