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 19th, 2013

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups

p-Nilpotent finite groups

• r "boaring" finite groups . . .

Definition Let p be a prime number. A finite group G is called p-nilpotent if G = P ⋉ Op′(G) for P ∈ Sylp(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 ∈ Sylp(G) one has NG(P) = P × Op′(NG(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 ∈ Syl2(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 ∈ Sylp(G) is abelian and NG(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 JR(P) = A ⊆ P | A abelian, rk(A) maximal , J0(P) = B ⊆ P | B abelian, |B| maximal . Theorem (J.G. Thompson (1964)) Let G be a finite group, let p be odd and let P ∈ Sylp(G). Then t.f.a.e.: G is p-nilpotent; CG(Z(P)) and NG(JR(P)) are p-nilpotent. J.G. Thompson

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 ∈ Sylp(G). Then t.f.a.e.: G is p-nilpotent; NG(Z(J0(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 ∈ Sylp(G). NG(P) p-nilpotent

???

G p-nilpotent. Example (a) Let V = V(p, Fp) and Affp(Fp) = GLp(Fp) ⋉ V. Put Cp = Z/pZ. Let p be odd, and let T◦ ⊂ GLp(Fp) be a maximal split torus, i.e., |T| = (p − 1)p. Then for G◦ = (Cp ⋉ T◦) ⋉ V one has for P ∈ Sylp(G) that NG◦(P) = P ≃ Cp ≀ Cp, but G◦ is not p-nilpotent. Let Tcox be a Coxeter torus, i.e., |Tcox| = pp − 1. Then for Gcox = (Cp ⋉ Tcox) ⋉ V and Q ∈ Sylp(Gcox) one has again that NGcox(Q) = Q ≃ Cp ≀ Cp, but Gcox is not p-nilpotent. (b) For G = GLn(F2) and P ∈ Syl2(G) one has that NG(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 ∈ Sylp(G), Q ≃ P, NG(Q) p-nilpotent = ⇒ G p-nilpotent. Example (W. Burnside (1911)) Abelian p-groups determine p-nilpotency locally. Notation By DNp 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 ζm(P), m ≥ 0, ζ0(P) = 1, denote the ascending central series of P, i.e., ζm+1(P)/ζm(P) = Z(P/ζm(P)); and Ω1(P) = g ∈ P | gp = 1 . Then P is called p-central of height k, k ≥ 1, if Ω1(P) ⊆ ζk(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 ∈ DNp.

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 Yp(1) = Cp ≀ Cp. For m ≥ 2 let Yp(m) be the p-group which is isomorphic to the pull-back of the diagram Cpm Cp Yp(m)

• Cp ≀ Cp

β

• where β : Cp ≀ Cp → Cp is the canonical map, and Cpm = Z/pmZ.

A p-group P is called slim, if ∀U ⊆ P ∀m ≥ 1 : U ≃ Yp(m). A p-group P is called xslim, if ∀U, V ⊆ P, V ⊳ U : U/V ≃ Yp(1). By slimp (resp. xslimp) we will denote the class of slim (resp. xslim) p-groups. In particular, xslimp slimp

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups

The Main Theorem

Theorem (T.W. (2012)) For p odd one has slimp ⊆ DNp. For p = 2 one has xslim2 ⊆ DN2. Theorem (T.W. (2012)) For p odd slimp ⊆ DNp is the maximal s-closed subclass of DNp. For p = 2 xslim2 ⊆ DN2 is the maximal s- and q-closed subclass

• f DN2.

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 ∈ xslimp. If P is of exponent p, then P ∈ xslimp. If P is regular (in the sense of P . Hall) then P ∈ xslimp. If srk(P) ≤ p − 1, where srk( ) is the sectional rank, then P ∈ xslimp. If p is odd and P is p-central of height p-1, then P ∈ slimp.

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) Fq[Cp]-module. Put G0 = Cp ⋉ Q. Let P0 be an irreducible, non-trivial (left) Fp[G0]-module. G = G0 ⋉ P0 will be called a pqp-sandwich group. Remark

• R. Brauer’s first Main Theorem implies that P0 is a projective

¯ Fp[G0]-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) ⊆ Op(Φ(G)). A p-Frattini extension π: X → G will be called a p-Schur-Frattini extension if ker(π) ∩ Op(X) ⊆ Z(Op(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 Yp(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 ∈ Sylp(G). Then G is p-nilpotent if, and

• nly if,

resG

P( ): H•(G, Fp) −

→ H•(P, Fp) 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 ∈ Sylp(G). Then G is p-nilpotent if, and only if, resG

P( ): H•(G, Fp) −

→ H•(P, Fp) is an F-isomorphism of rings.

• D. Quillen

Remark A homomorphism φ: A → B of finitely generated, graded commu- tative Fp-algebras is called an F-isomorphism if ker(φ) consists of nilpotent elements and for all b ∈ B there exists n ∈ N such that bpn ∈ 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 ∈ Sylp(G) the (injective) homomorphism of rings resG

NG(P)( ): H•(G, Fp) −

→ H•(Q, Fp)NG(Q). is an isomorphism. Theorem (R.G. Swan) Finite abelian p-groups are Swan groups. S.B. Priddy 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 ∈ Sylp(G), and H ⊆ G such that P ⊆ H ⊆ G. If for all A, B ⊆ P and g ∈ G such that ig|A : A → B is an isomorphism there exists h ∈ H such that ih|A = ig|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; NG(Q) controls p-fusion for every finite group G with P ≃ Q ∈ Sylp(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 = ⇒ NG(Q) controls p-fusion for all finite groups G with P ≃ Q ∈ Sylp(G). p odd: P p-central and finite = ⇒ P Swan. (J. Tate (1964)) p odd: P p-central and finite = ⇒ P ∈ DNp.

• J. Thévenaz

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups

Control of p-transfer

Definition Let G be a finite group, let P ∈ Sylp(G) and let H ⊆ G be such that P ⊆ H ⊆ G. The subgroup H is said to control p-transfer if trG,H : G/[G, G]Op(G) − → H/[H, H]Op(H) is an isomorphism.

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups

Yoshida’s theorem

Theorem (T. Yoshida (1978)) Let P be a finite p-group such that for all N ⊳ P one has P/N ≃ Yp(1) (where Yp(1) = Cp ≀ Cp). Then NG(Q) controls p-transfer for all finite groups G with P ≃ Q ∈ Sylp(G). Definition The finite p-group P will be said to be a Yoshida p-group if NG(Q) controls p-transfer for all finite subgroups G with P ≃ Q ∈ Sylp(G). Remark (G. Mislin (1990), J. Thévenaz (1993)) P Swan p-group = ⇒ P Yoshida p-group.

p-Nilpotency p-groups The Proof Swan groups Yoshida p-groups

The final picture

and thank you for your attention . . .

Swan

• Quillen

p=2

• C1
• DNp

Yoshida

G.Glauberman?

• slimp

p=2

• C2
• xslimp