Regular orbits

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Orbits of linear groups

Martin Liebeck

Imperial College London

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LetG ≤GLn(F) =GL(V), Fa field, G finite.

Will discuss results on orbits ofG onV:

1 Regular orbits

2 Number of orbits

3 Arithmetic conditions on orbit sizes, applications

Corresponding aﬃne permutation groupH :=VG ≤AGL(V), V = translation subgroup,G =H0 stabilizer of zero vector. Orbits ofG aresuborbits ofH, andHis primitive iﬀ G is irreducible onV.

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1 Regular orbits

2 Number of orbits

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1 Regular orbits

2 Number of orbits

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1 Regular orbits

2 Number of orbits

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1 Regular orbits

2 Number of orbits

Corresponding aﬃne permutation groupH:=VG ≤AGL(V), V = translation subgroup,G =H0 stabilizer of zero vector. Orbits ofG aresuborbits ofH, andHis primitive iﬀ G is irreducible onV.

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Regular orbits

LetG ≤GLn(F) =GL(V). ThenG has aregular orbit onV if∃ v∈V\0 such thatGv = 1. Regular orbit is v^G ={vg :g ∈G}, size|G|.

IfH=VG ≤AGL(V) is corresponding aﬃne permutation group onV, this saysH0v = 1, i.e. H has abase of size 2.

Do regular orbits exist?

Sometimes no: eg. ifG =GLn(q) (where F=F^q finite). In general, no regular orbit if|G| ≥ |V|.

Sometimes yes: eg. G =�s�,s a Singer cycle inGLn(q) of order qⁿ−1.

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Regular orbits

LetG ≤GLn(F) =GL(V). ThenG has a regular orbit onV if∃ v∈V\0 such thatGv = 1. Regular orbit is v^G ={vg :g ∈G}, size|G|.

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Regular orbits

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Regular orbits

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Regular orbits

Sometimes no: eg. ifG =GLn(q) (where F=F^q finite).

In general, no regular orbit if|G| ≥ |V|.

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Regular orbits

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Regular orbits

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Regular orbits

WhenFis infinite:

Lemma Let G ≤GLn(F) =GL(V), G finite, Finfinite. Then G has a regular orbit on V .

Proof Suppose false. ThenGv �= 1 ∀v ∈V. So everyv ∈V lies inCV(g) :={v ∈V :vg =v} for someg ∈G\1, ie.

V = �

g∈G\1

CV(g).

But asFis infinite, V is not a union of finitely many proper subspaces. Contradiction.

ForF finite this argument shows ∃regular orbit ifF finite and

|F|>|G|.

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Regular orbits

WhenFis infinite:

Lemma Let G ≤GLn(F) =GL(V), G finite,F infinite. Then G has a regular orbit on V .

V = �

g∈G\1

CV(g).

|F|>|G|.

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Regular orbits

WhenFis infinite:

Proof Suppose false. ThenGv �= 1 ∀v ∈V.

So every v ∈V lies inCV(g) :={v ∈V :vg =v} for someg ∈G\1, ie.

V = �

g∈G\1

CV(g).

|F|>|G|.

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Regular orbits

WhenFis infinite:

V = �

g∈G\1

CV(g).

|F|>|G|.

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Regular orbits

WhenFis infinite:

V = �

g∈G\1

CV(g).

|F|>|G|.

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Regular orbits

WhenFis infinite:

V = �

g∈G\1

CV(g).

|F|>|G|.

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Regular orbits

WhenFis infinite:

V = �

g∈G\1

CV(g).

ForF finite this argument shows∃ regular orbit ifF finite and

|F|>|G|.

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Regular orbits

WhenF=Fq finite,G ≤GLn(q) =GL(V):

Aim (i) If|G| ≥ |V|,G has no regular orbit onV

(ii) If|G|<|V|, proveG has a regular orbit, with the following exceptions....????

Far oﬀ. Delicate:

Example 1 Let G =Sc <GL_c−1(p) =GL(V), where p >c and V ={(a1, . . . ,ac) :ai ∈F^p,�

ai = 0}. Then G has regular orbits onV. Number of regular orbits is

1

c!(p−1)(p−2)· · ·(p−c+ 1).

Example 2 Let G =Sc×C2 <GLc−1(p) =GL(V), where p=c+ 1, V as above, C2 =�−1V�. ThenG has no regular orbits onV.

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Regular orbits

Far oﬀ. Delicate:

ai = 0}. Then G has regular orbits onV. Number of regular orbits is

1

c!(p−1)(p−2)· · ·(p−c+ 1).

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Regular orbits

Far oﬀ. Delicate:

ai = 0}. ThenG has regular orbits onV. Number of regular orbits is

1

c!(p−1)(p−2)· · ·(p−c+ 1).

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Regular orbits

Far oﬀ. Delicate:

ai = 0}. ThenG has regular orbits onV. Number of regular orbits is

1

c!(p−1)(p−2)· · ·(p−c+ 1).

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More on egular orbits

In general letG <GLn(q) with q varying, but fixed Brauer

character ofG. There is a poly f(x) of degree n such that number of regular orbits ofG is f(q). So ∃regular orbits for all but at mostn values ofq.

Eg. G =Sc <GL_c−1(q), char(Fq)>n: here

f(q) = _c!¹(q−1)(q−2)· · ·(q−c+ 1). This is a poly in q with roots equal to the exponents of the Weyl groupW(Ac−1)∼=Sc. Same holds for all finite reflection groups in their natural

representations (Orlik-Solomon). Eg. forG =W(F4)<GL4(q), number of regular orbits

1

|W(F4)|(q−1)(q−5)(q−7)(q−11).

Not always so nice, eg. forG =PSL2(7)<GL3(q) (of index 2 in a unitary reflection group),f(q) = ₁₆₈¹ (q−1)(q²+q−48).

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More on egular orbits

character ofG. There is a poly f(x) of degree n such that number of regular orbits ofG is f(q).

So ∃regular orbits for all but at mostn values ofq.

1

|W(F4)|(q−1)(q−5)(q−7)(q−11).

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More on egular orbits

1

|W(F4)|(q−1)(q−5)(q−7)(q−11).

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More on egular orbits

Eg. G =Sc <GLc−1(q), char(Fq)>n: here

f(q) = _c!¹(q−1)(q−2)· · ·(q−c+ 1). This is a poly in q with roots equal to the exponents of the Weyl groupW(Ac−1)∼=Sc.

Same holds for all finite reflection groups in their natural

1

|W(F4)|(q−1)(q−5)(q−7)(q−11).

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More on egular orbits

1

|W(F4)|(q−1)(q−5)(q−7)(q−11).

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More on egular orbits

1

|W(F4)|(q−1)(q−5)(q−7)(q−11).

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Still more on regular orbits

General theory (Pahlings-Plesken): G <GLn(q) with q varying, fixed Brauer character. For each subgroupH <G there is a poly fH(q) of degree dimCV(H) such that number of orbits ofG with stabilizer conjugate toH isfH(q). ∃methods for computing these polys using “table of marks” ofG.

Eg. here’s the table of marks forA5: A5/C1 60

A5/C2 30 2 A5/C3 20 2 A5/V4 15 3 3

A5/C5 12 2

A5/S3 10 2 1 1

A5/D10 6 2 1 1

A5/A4 5 1 2 1 1

A5/A5 1 1 1 1 1 1 1 1 1

C1 C2 C3 V4 C5 S3 D10 A4 A5

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Still more on regular orbits

General theory (Pahlings-Plesken): G <GLn(q) with q varying, fixed Brauer character. For each subgroupH <G there is a poly fH(q) of degree dimCV(H) such that number of orbits ofG with stabilizer conjugate toH isfH(q). ∃ methods for computing these polys using “table of marks” ofG.

A5/C2 30 2 A5/C3 20 2 A5/V4 15 3 3

A5/C5 12 2

A5/S3 10 2 1 1

A5/D10 6 2 1 1

A5/A4 5 1 2 1 1

A5/A5 1 1 1 1 1 1 1 1 1

C1 C2 C3 V4 C5 S3 D10 A4 A5

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Still more on regular orbits

General theory (Pahlings-Plesken): G <GLn(q) with q varying, fixed Brauer character. For each subgroupH <G there is a poly fH(q) of degree dimCV(H) such that number of orbits ofG with stabilizer conjugate toH isfH(q). ∃ methods for computing these polys using “table of marks” ofG.

A5/C2 30 2 A5/C3 20 2 A5/V4 15 3 3

A5/C5 12 2

A5/S3 10 2 1 1

A5/D10 6 2 1 1

A5/A4 5 1 2 1 1

A5/A5 1 1 1 1 1 1 1 1 1

C1 C2 C3 V4 C5 S3 D10 A4 A5

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Yet more on regular orbits

Regular orbits ofG <GLn(q) crop up in several areas. A couple of examples:

1. Supposeallorbits regular, ie. Gv = 1∀v ∈V\0. Then aﬃne groupH=VG ≤AGL(V) is a Frobenius group(Hvw = 1∀v,w) andG aFrobenius complement. Classified by Zassenhaus. Eg. SL2(5)<GL2(q) (char >5)

OrSL2(5)⊗(Cr.Cs)<GL2(q)⊗GLr(q)<GL2r(q) (r|s−1, r,s >5)

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Yet more on regular orbits

1. Supposeallorbits regular, ie. Gv = 1∀v ∈V\0. Then aﬃne groupH=VG ≤AGL(V) is a Frobenius group(Hvw = 1∀v,w) andG aFrobenius complement. Classified by Zassenhaus. Eg. SL2(5)<GL2(q) (char >5)

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Yet more on regular orbits

1. Supposeallorbits regular, ie. Gv = 1∀v ∈V\0. Then aﬃne groupH=VG ≤AGL(V) is a Frobenius group(Hvw = 1∀v,w) andG aFrobenius complement. Classified by Zassenhaus.

Eg. SL2(5)<GL2(q) (char >5)

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Yet more on regular orbits

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Yet more on regular orbits

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Yet more on regular orbits

2. The k(GV)-problem This is

Conjecture Let G <GLn(p) =GL(V), G a p^�-group. The number of conjugacy classes k(VG) in the semidirect product VG satisfies

k(VG)≤ |V|.

Equivalent top-soluble case of Brauer’sk(B)-problem. Equality can hold, eg. G =�Singer−cycle� ∼=Cpⁿ−1

Robinson-Thompson reduction: conjecture proved if we can show that forG of “simple type” or ”extraspecial type”, there exists a regular orbit ofG on V.

“Simple type”: G has an irreducible normal subgroup H such that H/Z(H) is non-abelian simple.

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Yet more on regular orbits

k(VG)≤ |V|.

Equivalent top-soluble case of Brauer’sk(B)-problem. Equality can hold, eg. G =�Singer−cycle� ∼=Cpⁿ−1

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Yet more on regular orbits

k(VG)≤ |V|.

Equivalent top-soluble case of Brauer’sk(B)-problem.

Equality can hold, eg. G =�Singer−cycle� ∼=Cpⁿ−1

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Yet more on regular orbits

k(VG)≤ |V|.

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Yet more on regular orbits

k(VG)≤ |V|.

Robinson-Thompson reduction: conjecture proved if we can show that forG of “simple type” or ”extraspecial type”, there exists a regular orbit ofG onV.

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Yet more on regular orbits

k(VG)≤ |V|.

Robinson-Thompson reduction: conjecture proved if we can show that forG of “simple type” or ”extraspecial type”, there exists a regular orbit ofG onV.

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Last one on regular orbits

Theorem (Hall-L-Seitz, Goodwin, Kohler-Pahlings, Riese) If G <GLn(p) is a p^�-group of simple type, then G has a regular orbit unless one of:

(i) Ac�G <GLc−1(p), p >c

(ii) 23 exceptional cases, all with n≤10,p ≤61.

Eventuallyk(GV)-conjecture proved (Gluck, Magaard, Riese, Schmidt 2004)

General classification of linear groups with/without regular orbits is out of reach at the moment. Need substitutes...

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Last one on regular orbits

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Last one on regular orbits

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Last one on regular orbits

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Few orbits

LetG ≤GLn(q) =GL(V). There are results classifying groups G with few orbits onV\0.

One orbit G transitive on V\0. Equivalently, aﬃne group VG ≤AGL(V) is 2-transitive.

Hering’s theorem Classification of transitive linear groups G ≤GLn(q):

(i)G ≥SLn(q),Spn(q)

(ii)G ≥G2(q) (n= 6, q even) (iii)G ≤ΓL1(qⁿ)

(iv)∼10 exceptions, all with |V| ≤59² (eg. F^∗59◦SL2(5)<GL2(59)).

Two orbits ∃ similar classification (L) – hence the rank 3 aﬃne permutation groups.

Three, four,... orbits: can be done if desperate

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Few orbits

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Few orbits

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Few orbits

(iv)∼10 exceptions, all with |V| ≤59² (eg.

F^∗59◦SL2(5)<GL2(59)).

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Few orbits

F^∗59◦SL2(5)<GL2(59)).

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Few orbits

F^∗59◦SL2(5)<GL2(59)).

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Arithmetic conditions on orbit sizes

Half-transitivity G ≤GLn(q) =GL(V) is ¹₂-transitive if all orbits ofG onV\0 have equal size. (Aﬃne groupVG ≤AGL(V) is then

3

2-transitive.) Many examples, eg.

(a)G transitive

(b)G ≤ �s�,s Singer cycle of order qⁿ−1

(c)G a Frobenius complement, eg. SL2(5)<GL2(q) (d)G =S(q)<GL2(q) (q odd), where

S(q) ={

� a 0 0 ±a⁻¹

� ,

� 0 a

±a⁻¹ 0

�

:a∈F^∗q} Passman 1969The soluble ¹₂-transitive linear groups are: Frobenius complements, subgroups of ΓL1(qⁿ), S(q), and 6 exceptions withqⁿ≤17².

General case....???

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Arithmetic conditions on orbit sizes

Half-transitivity G ≤GLn(q) =GL(V) is ¹₂-transitive if all orbits ofG on V\0 have equal size. (Aﬃne groupVG ≤AGL(V) is then

3

2-transitive.)

Many examples, eg. (a)G transitive

S(q) ={

� a 0 0 ±a⁻¹

� ,

� 0 a

±a⁻¹ 0

�

General case....???

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Arithmetic conditions on orbit sizes

3

(a)G transitive

S(q) ={

� a 0 0 ±a⁻¹

� ,

� 0 a

±a⁻¹ 0

�

General case....???

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Arithmetic conditions on orbit sizes

3

(a) G transitive

S(q) ={

� a 0 0 ±a⁻¹

� ,

� 0 a

±a⁻¹ 0

�

General case....???

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Arithmetic conditions on orbit sizes

3

(a) G transitive

(b) G ≤ �s�,s Singer cycle of orderqⁿ−1

S(q) ={

� a 0 0 ±a⁻¹

� ,

� 0 a

±a⁻¹ 0

�

General case....???

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Arithmetic conditions on orbit sizes

3

(a) G transitive

(c) G a Frobenius complement, eg. SL2(5)<GL2(q)

(d)G =S(q)<GL2(q) (q odd), where S(q) ={

� a 0 0 ±a⁻¹

� ,

� 0 a

±a⁻¹ 0

�

General case....???

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Arithmetic conditions on orbit sizes

3

(a) G transitive

(c) G a Frobenius complement, eg. SL2(5)<GL2(q) (d) G =S(q)<GL2(q) (q odd), where

S(q) ={

� a 0 0 ±a⁻¹

� ,

� 0 a

±a⁻¹ 0

�

:a∈F^∗q}

Passman 1969The soluble ¹₂-transitive linear groups are: Frobenius complements, subgroups of ΓL1(qⁿ), S(q), and 6 exceptions withqⁿ≤17².

General case....???

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Arithmetic conditions on orbit sizes

3

(a) G transitive

S(q) ={

� a 0 0 ±a⁻¹

� ,

� 0 a

±a⁻¹ 0

�

:a∈F^∗q} Passman 1969The soluble ¹₂-transitive linear groups are:

Frobenius complements, subgroups of ΓL1(qⁿ),S(q), and 6 exceptions withqⁿ≤17².

General case....???

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Arithmetic conditions on orbit sizes

3

(a) G transitive

S(q) ={

� a 0 0 ±a⁻¹

� ,

� 0 a

±a⁻¹ 0

�

:a∈F^∗q} Passman 1969The soluble ¹₂-transitive linear groups are:

Frobenius complements, subgroups of ΓL1(qⁿ),S(q), and 6 exceptions withqⁿ≤17².

General case....???

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Arithmetic conditions

p-exceptional groups Say G ≤GLn(p) =GL(V) is p-exceptional ifp divides G and all orbits ofG onV have size coprime to p. Ties in with previous notions:

(a)G p-exceptional ⇒ G hasno regular orbit onV (b)G transitive,p| |G| ⇒G p-exceptional

(c)G ¹₂-transitive,p| |G| ⇒ G p-exceptional

∃many examples, eg. V =W^k,G =H wrK where H≤GL(W) is transitive onW\0 and K ≤Sk has all orbits on the power set of {1, . . . ,k}of p^�-size. (Eg. K ap^�-group)

Canp-exceptional linear groups be classified?

Yes, at least the irreducible ones ((Giudici, L, Praeger, Saxl, Tiep)...

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Arithmetic conditions

p-exceptional groups Say G ≤GLn(p) =GL(V) is p-exceptional ifp divides G and all orbits ofG onV have size coprime to p.

Ties in with previous notions:

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Arithmetic conditions

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Arithmetic conditions

(a)G p-exceptional ⇒ G hasno regular orbit onV

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Arithmetic conditions

(a)G p-exceptional ⇒ G hasno regular orbit onV (b)G transitive, p| |G| ⇒G p-exceptional

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Arithmetic conditions

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Arithmetic conditions

∃many examples, eg. V =W^k,G =H wrK where H≤GL(W) is transitive onW\0 andK ≤Sk has all orbits on the power set of {1, . . . ,k}of p^�-size. (Eg. K ap^�-group)

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Arithmetic conditions

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Arithmetic conditions

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Arithmetic conditions

(G ≤GLn(p) =GL(V) isp-exceptionalifp dividesG and all orbits ofG onV have size coprime to p.)

Theorem Let G ≤GLd(p) =GL(V) be p-exceptional. Suppose G acts irreducibly and primitively on V . Then one of:

(i) G transitive on V\0 (ii) G ≤ΓL1(pⁿ)

(iii) G =Ac,Sc <GLc−�(2), c = 2^r −1 or 2^r −2,�= 1 or 2 (iv) G^� =SL2(5)<GL4(3), orbits1,40,40

PSL2(11)<GL5(3), orbits1,22,110,110 M11<GL5(3), orbits1,22,220

M23<GL11(2), orbits 1,23,253,1771

Also have a classification of the imprimitivep-exceptional groups: V =W^k,G ≤HwrK whereH≤GL(W) is transitive onW\0 andK ≤Sk has all orbits on the power set of {1, . . . ,k}of p^�-size.

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Arithmetic conditions

(iii) G =Ac,Sc <GLc−�(2), c = 2^r −1 or2^r −2,�= 1 or 2 (iv) G^� =SL2(5)<GL4(3), orbits1,40,40

M23<GL11(2), orbits 1,23,253,1771

Also have a classification of the imprimitivep-exceptional groups: V =W^k,G ≤HwrK whereH≤GL(W) is transitive onW\0 andK ≤Sk has all orbits on the power set of {1, . . . ,k}of p^�-size.

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Arithmetic conditions

(iii) G =Ac,Sc <GLc−�(2), c = 2^r −1 or2^r −2,�= 1 or 2 (iv) G^� =SL2(5)<GL4(3), orbits1,40,40

M23<GL11(2), orbits 1,23,253,1771

Also have a classification of the imprimitivep-exceptional groups:

V =W^k,G ≤HwrK whereH≤GL(W) is transitive onW\0 andK ≤Sk has all orbits on the power set of {1, . . . ,k}of p^�-size.

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Consequences

RecallG ¹₂-transitive ⇒all orbits on V\0 have same size⇒ G is p-exceptional. Hence

Theorem If G ≤GLd(p) is ¹₂-transitive and p divides |G|, then one of:

(i) G is transitive on V\0 (ii) G ≤ΓL1(p^d)

(iii) G^� =SL2(5)<GL4(3), orbits1,40,40.

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Consequences

(iii) G^� =SL2(5)<GL4(3), orbits1,40,40.

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Consequences

(iii) G^� =SL2(5)<GL4(3), orbits1,40,40.

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Consequences

Gluck-Wolf theorem 1984 Let p be a prime and G a finite p-soluble group. Suppose N�G and N has an irreducible character φsuch thatχ(1)/φ(1) is coprime to p for allχ⊆φ^G. Then G/N has abelian Sylow p-subgroups.

This implies Brauer’sheight zeroconjecture forp-soluble groups. Using our classification ofp-exceptional groups, Tiep and Navarro have proved the Gluck-Wolf theorem for arbitrary finite groupsG. May lead to the complete solution of the height zero conjecture.

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Consequences

This implies Brauer’sheight zeroconjecture forp-soluble groups. Using our classification ofp-exceptional groups, Tiep and Navarro have proved the Gluck-Wolf theorem for arbitrary finite groupsG. May lead to the complete solution of the height zero conjecture.

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Consequences

This implies Brauer’sheight zeroconjecture forp-soluble groups.

Using our classification ofp-exceptional groups, Tiep and Navarro have proved the Gluck-Wolf theorem for arbitrary finite groupsG. May lead to the complete solution of the height zero conjecture.

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Consequences

This implies Brauer’sheight zeroconjecture forp-soluble groups.

Using our classification ofp-exceptional groups, Tiep and Navarro have proved the Gluck-Wolf theorem for arbitrary finite groupsG. May lead to the complete solution of the height zero conjecture.