nonsolvable groups of which the smith sets are groups groups of which the smith sets are groups...

Nonsolvable groups of which the Smith sets aregroups

Toshio Sumi

November 19, 2011

The 38th Symposium on Transformation Groups

Toshio Sumi (November 19, 2011) Nonsolvable groups 38th Symp. on Transf. Gr. 1 / 28

Smith set

Smith Equivalence

Two real G-modules U and V are called Smith equivalent if there existsa smooth action of G on a homotopy sphere ! such that !G = {x, y} fortwo points x and y at which Tx(!) ! U and Ty(!) ! V as real G-modules.Let Sm(G) be a subset of the real representation ring RO(G) of Gconsisiting of the differences U ! V of real G-modules U and V whichare Smith equivalent.If !P is connected for a subgroup P of G of prime power order, then wecall that U and V which are c-primary Smith equivalent. Let PSmc(G)be the subset of Sm(G) consisting the differences of real G-moduleswhich are c-primary Smith equivalent and the zero element forconvenience.


Smith set

Notations

RO(G) denotes the real representation ring.P(G) denotes the set of all subgroups of G of prime power orderand {1}.Op(G) denotes the Dress subgroup of type p for a prime p.

Op(G) =!

L!G;[G:L]=paL

L(G) denotes the set of all subgroups L of G such that L " Op(G)for some prime p.Gnil denotes the smallest normal subgroup of G by which quotientis nilpotient.

Gnil =!

pOp(G)


Smith set

Notations

For subsets F1 and F2 of subgroups of G and a subset A of RO(G),

AF1 =!

P#F1ker(ResGP : RO(G)$ RO(P)) %A

AF2 =!

L#F2ker(FixL : RO(G)$ RO(NG(L)/L)) %A

AF2F1 = AF1 %AF2 =

!

P#F1ker ResGP %

!

L#F2ker FixL %A


Smith set

Notations

For subsets F1 and F2 of subgroups of G and a subset A of RO(G),

AF1 =!

P#F1ker(ResGP : RO(G)$ RO(P)) %A

AF2 =!

L#F2ker(FixL : RO(G)$ RO(NG(L)/L)) %A

AF2F1 = AF1 %AF2 =

!

P#F1ker ResGP %

!

L#F2ker FixL %A

Remark

If A is a subgroup of RO(G) then AF1 , AF2 , and AF2F1 are alsosubgroups.


Smith set

Upper bound for Sm(G)Since the fixed points are isolated,

Sm(G) & RO(G){G}.

PutN2(G) := {H ! G | [G : N] ' 2} and

%N2(G) :=!

H#N2(G)H.

%N2(G) " Gnil and RO(G){%N2(G)} = RO(G)N2(G).

Morimoto pointed out that

Sm(G) & RO(G)N2(G)

which impliesSm(Aut(A6)) = {0}.


Smith set

If Sm(G) is a group, then ...

PSmc(G) &!!!!!$ RO(G)N2(G)P(G)&!!!!!$ RO(G)P(G)

%"""""# %

"""""#

Sm(G) &!!!!!$ RO(G)N2(G){P#P(G)||P| =odd or 1,2,4}

RemarkIf Sm(G) is a group, then

Sm(G) & RO(G)N2(G)P(G) .

If Sm(G) is not a subset of RO(G)P(G), then Sm(G) is not a group.


Smith Equivalence Question


(g) for g # G denotes the conjugacy class of g in G.




(g) for g # G denotes the conjugacy class of g in G.(g)± = (g) ( (g!1): real conjugacy class




(g) for g # G denotes the conjugacy class of g in G.(g)± = (g) ( (g!1): real conjugacy classThe number of irreducible real G-modules is equal to one of allreal conjugacy classes in G.




(g) for g # G denotes the conjugacy class of g in G.(g)± = (g) ( (g!1): real conjugacy classThe number of irreducible real G-modules is equal to one of allreal conjugacy classes in G.rG denotes the number of real conjugacy classes of elements of Gnot of prime power order.




(g) for g # G denotes the conjugacy class of g in G.(g)± = (g) ( (g!1): real conjugacy classThe number of irreducible real G-modules is equal to one of allreal conjugacy classes in G.rG denotes the number of real conjugacy classes of elements of Gnot of prime power order.

rankRO(G)P(G) = rGrankRO(G)N2(G)P(G) ' rankRO(G)

{G}P(G) = rG ! 1

Laitinen and Pawa!owski (1999) obtained that if aG ' 1 thenPSmc(G) = {0}. If Sm(G) is a group and rankRO(G)N2(G)P(G) = 0 then

Sm(G) = {0}.



Non-solvable groups

Let G be a non-solvable group.

TheoremPSmc(G) " {0} if and only if rG " 2 and G ! Aut(A6).

PSmc(G) = {0} if rG ' 1. (Laitinen-Pawa!owski,1999)PSmc(G) " {0} if G is a gap group with rG " 2 andG ! Aut(A6), P!L(2, 27). (Pawa!owski-Solomon, 2002)PSmc(Aut(A6)) = {0} (Morimoto, 2008)PSmc(P!L(2, 27)) = Z (Morimoto, 2010)PSmc(G) " {0} if rG " 2 and G ! Aut(A6), P!L(2, 27).(Pawa!owski-Sumi, to appear)

1$ PS L(2, 27)$ P!L(2, 27)$ Aut(F27)$ 1



Non-solvable groups

How about is for Sm(G)? It is unknown whether Sm(G) = {0} for thefollowing non-solvable group G with rG ' 1.

G/F(G) ! S 5, S L(2, 8), S z(8) or S z(32) where F(G) is a non-identityelementary abelian 2-group and CF(G)(x) = 1 for every x # G of oddorder.



Group Problem

ProblemFind a finite group G such that Sm(G) is a group.Find a finite group G such that PSmc(G) is not a group if exists.Find a finite group G such that PSmc(G) " Sm(G)P(G) if exists.

LemmaIf Sm(G) is a group then Sm(G) = Sm(G)P(G).



Group Problem

Theorem (Cappell-Shanesson)Sm(Z/4n) " {0} for n " 2.

Therefore, Sm(K ) Z/4n) is not a group for n " 2.

To find a finite group G such that Sm(G) is a group, we must studyWhen does Sm(G) = Sm(G)P(G) hold?What is difference between PSmc(G) and Sm(G)P(G)?


Connectivity

When Sm(G) = Sm(G)P(G)?

Pe(G) = {P # P(G) | |P| = 2a, a " 3}Elme(G) = {g # G | *g+ # Pe(G)}

Proposition (Pawalowski-Sumi, 2009)

If g*g2+ & (g)± for g # Elme(G), then it holds that Sm(G)P(G) = Sm(G).

ExampleAn and S n.

0Pawa!owski, K. and Sumi, T., The Laitinen conjecture for finite solvableOliver groups, Proc. Amer. Math. Soc., 2009, 2147–2156.


Connectivity


Proposition

PSmc(An) = Sm(An) = RO(An){An}P(An)which is a free abelian group of rank max(rAn ! 1, 0), and

PSmc(S n) = Sm(S n) = RO(S n){An}P(S n)

which is a free abelian group of rank$%%%%%&%%%%%'

0, n = 2, 3, 4, 5,1, n = 6,rS n ! 2 (" 3), n " 7.


Connectivity

When Sm(G) = Sm(G)P(G)?Let Irr(G) be the set of representatives of isomorphism classes of theirreducible real G-modules and put

iR(g,G) =((({V # Irr(G) | dimVg = 0 = dimV%N2(G)}

(((

for g # G and

iR(G) = max ({iR(g,G) | g # Elme(G)} ( {0}) .

Proposition (Pawalowski-Sumi, 2009 (generalized version))For a group G and its quotient group K, it holds that iR(K) ' iR(G). IfiR(G) ' 1 then Sm(G)P(G) = Sm(G).

Example

PS L(2, q), PS L(3, q), PGL(2, q), PGL(3, q), PSU(3, q2), sporadic groups,automorphism groups of sporadic groups.


Connectivity


ExampleiR(G) ' 1 for the following groups G:PS L(2, q), PS L(3, q), PGL(2, q), PGL(3, q), PSU(3, q2), sporadic groups,automorphism groups of sporadic groups.

PropositionFor any groups G listed in the above example,

Sm(G) = Sm(G)P(G).


Connectivity


ExampleiR(G) ' 1 for the following groups G:PS L(2, q), PS L(3, q), PGL(2, q), PGL(3, q), PSU(3, q2), sporadic groups,automorphism groups of sporadic groups.

PropositionFor any groups G listed in the above example,

Sm(G) = Sm(G)P(G).

RemarkThere still exist (infinitely) many groups for which is unknown whether

Sm(G) = Sm(G)P(G).


Connectivity

Oliver group

A group G is an Oliver group if there does not exist a sequenceP " H "G of subgroups of G such that P and G/H are groups of primepower order and H/P is cyclic.


Connectivity

Oliver group

A group G is an Oliver group if there does not exist a sequenceP " H "G of subgroups of G such that P and G/H are groups of primepower order and H/P is cyclic.

Theorem (Oliver, 1996, Laitinen-Morimoto, 1998)The following claims are equivalent.

G is an Oliver group.There exists a fixed point free action on a disk.There exists a one fixed point action on a sphere.

Oliver, B., Fixed point sets and tangent bundles of actions on disks andEuclidean spaces, Topology, 35, 1996, 583–615.

Laitinen, E. and Morimoto, M., Finite groups with smooth one fixed pointactions on spheres, Forum Math., 10, 1998, 479–520.


Connectivity

Nil-P-condition

Theorem (Morimoto)

Let G be an Oliver group with [G : Gnil] = 3. It occurs thatSm(G) & RO(G)Gnil , for example if a Sylow 2-subgroup of G is normal,and also occurs that Sm(G) # RO(G)Gnil , for example G = P!L(2, 27).


Connectivity

Nil-P-condition

Theorem (Morimoto)

Let G be an Oliver group with [G : Gnil] = 3. It occurs thatSm(G) & RO(G)Gnil , for example if a Sylow 2-subgroup of G is normal,and also occurs that Sm(G) # RO(G)Gnil , for example G = P!L(2, 27).

DefinitionFor a normal subgroup N of G, we say that G satisfies theN-P-condition if there are real G-modules U and V such thatUN = VN = 0 and [R , U] ! [V] # RO(G)P(G). If N = Gnil we say that Gsatisfies the Nil-P-condition.

If a Sylow 2-subgroup of G is normal, G does not satisfy theNil-P-condition.


Connectivity

Nil-P-condition

DefinitionFor a normal subgroup N of G, we say that G satisfies theN-P-condition if there are real G-modules U and V such thatUN = VN = 0 and [R , U] ! [V] # RO(G)P(G).

PropositionLet f : G $ G- be a epimorphism and let N be a normal subgroup of G.If G- satisfies the N-P-condition then G satisfies the N-P-condition.


Connectivity

Nil-P-condition


PropositionLet N be a normal subgroup of G. If there are a subgroup K of G andan epimorphism f : K $ H such that f (K % N) = H, KN = G and H hassubquotient isomorphic to D2pq, where p and q are distinct primes,then G satisfies the N-P-condition for any subgroup L ' G with L " N,


Connectivity

Main Theorem

TheoremLet G be a finite group possessing a quotient group which is a gapOliver group satisfying the Nil-P-condition. Then

PSmc(G) = RO(G)N2(G)P(G)

and PSmc(G) is a group of rank

rankRO(G/Gnil)N2(G/Gnil) + rankRO(G){G

nil}P(G) .


Connectivity

Proposition (cf. Laitinen-Pawa!owski, 1999)For a perfect group G, it holds that

PSmc(G) = RO(G){G}P(G).

By Oliver, for a perfect group, it satisfies the Nil-P-condition if and onlyif there is a subquotient group isomorphic to a dihedral group D2pq oforder 2pq, where p and q are distinct primes.

TheoremLet m " 3 be integer not of a power of 2 and n an integer not of primepower. Let K be a finite group such that a Sylow 2-subgroup of K is nota subgroup of a product of dihedral groups. It holds that


for G = K ) (D2n)Cm).


Connectivity

Nil-P-condition

TheoremLet q > 1 be a prime power. The following gap groups satisfy theNil-P-condition.

1 Symmetric groups S n, n " 7


Connectivity

Nil-P-condition


1 Symmetric groups S n, n " 72 Projective general linear groups PGL(2, q), q " 2, 3, 4, 5, 7, 8, 9, 17


Connectivity

Nil-P-condition


1 Symmetric groups S n, n " 72 Projective general linear groups PGL(2, q), q " 2, 3, 4, 5, 7, 8, 9, 173 Projective general linear groups PGL(3, q), q " 2, 4, 8


Connectivity

Nil-P-condition


1 Symmetric groups S n, n " 72 Projective general linear groups PGL(2, q), q " 2, 3, 4, 5, 7, 8, 9, 173 Projective general linear groups PGL(3, q), q " 2, 4, 84 Projective general linear groups PGL(n, q), n " 4


Connectivity

Nil-P-condition


1 Symmetric groups S n, n " 72 Projective general linear groups PGL(2, q), q " 2, 3, 4, 5, 7, 8, 9, 173 Projective general linear groups PGL(3, q), q " 2, 4, 84 Projective general linear groups PGL(n, q), n " 45 General linear groups GL(2, q), q " 2, 3, 4, 5, 7, 8, 9, 176 General linear groups GL(3, q), q " 2, 4, 87 General linear groups GL(n, q), n " 4


Connectivity

Nil-P-condition


1 Symmetric groups S n, n " 72 Projective general linear groups PGL(2, q), q " 2, 3, 4, 5, 7, 8, 9, 173 Projective general linear groups PGL(3, q), q " 2, 4, 84 Projective general linear groups PGL(n, q), n " 45 General linear groups GL(2, q), q " 2, 3, 4, 5, 7, 8, 9, 176 General linear groups GL(3, q), q " 2, 4, 87 General linear groups GL(n, q), n " 48 The automorphism group of sporadic groups


Connectivity

Nil-P-condition


PropositionLet N be a normal subgroup of G. If there are a subgroup K of G andan epimorphism f : K $ H such that f (K % N) = H, KN = G and H hassubquotient isomorphic to D2pq, where p and q are distinct primes,then G satisfies the N-P-condition for any subgroup L ' G with L " N,


Connectivity

Nil-P-condition


1 Alternating groups An, n " 72 Projective special linear groups PS L(2, q), q " 2, 3, 4, 5, 7, 8, 9, 173 Projective special linear groups PS L(3, q), q " 2, 4, 84 Projective special linear groups PS L(n, q), n " 45 Special linear groups S L(2, q), q " 2, 3, 4, 5, 7, 8, 9, 176 Special linear groups S L(3, q), q " 2, 4, 87 Special linear groups S L(n, q), n " 48 Sporadic groups


Outline of the proof

Main Theorem

TheoremLet G be a finite group possessing a quotient group which is a gapOliver group satisfying the Nil-P-condition. Then


and PSmc(G) is a group of rank


nil}P(G) .




Theorem (Morimoto, 2010)Let G be an Oliver group and V a gap real G-module. Let (U1,V1),(U2,V2) and (W1,W2) be P(G)-matched pairs of real G-modules suchthat U%N2(G)1 = R = U%N2(G)2 , VGnil1 = 0 = VGnil2 , Wj = (Uj ! R),mj , Xj forj = 1, 2, where mj is a nonnegative integer and Xj is an L(G)-free realG-module. Then there exist positive integers N1 and N2 such that forany integers a " N1 and b " N2, one has a smooth G-action on astandard sphere S having the following properties.

1 SG = {y1, y2}.2 Tyj(S ) ! Wj , V,a , R[G],bL(G) for j = 1, 2.3 dim S H " 6 for all H # S (G) "L(G).

Nontrivial P(G)-matched S-related Pairs for Finite Gap Oliver Groups, J.Japan Math. Soc., 62, 2010, 623–647.




Putting X1 = X2 = 0 and m1 = m2 = 1 in this theorem, we obtain

TheoremLet G be a gap Oliver group. Let (U1,U2) be a P(G)-matched pair ofreal G-modules such that U%N2(G)1 = R = U%N2(G)2 . If there existG-modules V1 and V2 such that

(Uj,Vj) is a P(G)-matched pair for j = 1, 2 and

VGnil1 = 0 = VGnil2 ,then [U1] ! [U2] # PSmc(G).





(Uj,Vj) is a P(G)-matched pair for j = 1, 2 and VGnil1 = 0 = VGnil2 ,then [U1] ! [U2] # PSmc(G).





(Uj,Vj) is a P(G)-matched pair for j = 1, 2 and VGnil1 = 0 = VGnil2 ,then [U1] ! [U2] # PSmc(G).

TheoremLet G be a gap Oliver group. Suppose that for any # # Irr(G/Gnil) thereis an element X# # RO(G){G

nil} such that [#] + X# # RO(G)P(G). ThenPSmc(G) = RO(G)N2(G)P(G) and PSmc(G) is a group of rank


nil}P(G) .


Future work

Future work

Problem

Sm(A5 )Cn4) =?, Sm(A5 )Cn

8) =?

PSmc(A5 )C4) = Sm(A5 )C4) ! Z4

ButPSmc(A5 )C8) =? < Sm(A5 )C8) =?


nonsolvable groups of which the smith sets are groups groups of which the smith sets are groups...

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