Torsion Units in Integral Group Rings Andreas B achle Vrije - - PowerPoint PPT Presentation

Torsion Units in Integral Group Rings

Andreas B¨ achle

Vrije Universiteit Brussel

Computational Methods for Representations and Group Rings

University of Stuttgart, February 18 and 19, 2016

Notations

G finite group

Notations

G finite group ZG integral group ring of G

Notations

G finite group ZG integral group ring of G ε augmentation map of ZG, i.e. ε

• g∈G

rgg

• =

g∈G

rg.

Notations

G finite group ZG integral group ring of G ε augmentation map of ZG, i.e. ε

• g∈G

rgg

• =

g∈G

rg. U(ZG) group of units of ZG

Notations

G finite group ZG integral group ring of G ε augmentation map of ZG, i.e. ε

• g∈G

rgg

• =

g∈G

rg. U(ZG) group of units of ZG V(ZG) group of units of ZG of augmentation 1 aka normalized units.

Notations

G finite group ZG integral group ring of G ε augmentation map of ZG, i.e. ε

• g∈G

rgg

• =

g∈G

rg. U(ZG) group of units of ZG V(ZG) group of units of ZG of augmentation 1 aka normalized units. U(ZG) = ±V(ZG).

Questions

How strong is the connection between the group G and the set of torsion subgroups in V(ZG)?

Questions

How strong is the connection between the group G and the set of torsion subgroups in V(ZG)? For example, for finite X ≤ V(ZG) we have

◮ |X|

• |G|

(ˇ Zmud, Kurenno˘ ı 1967)

Questions

How strong is the connection between the group G and the set of torsion subgroups in V(ZG)? For example, for finite X ≤ V(ZG) we have

◮ |X|

• |G|

(ˇ Zmud, Kurenno˘ ı 1967)

◮ exp X

• exp G

(Cohn, Livingstone, 1965)

Questions

How strong is the connection between the group G and the set of torsion subgroups in V(ZG)? For example, for finite X ≤ V(ZG) we have

◮ |X|

• |G|

(ˇ Zmud, Kurenno˘ ı 1967)

◮ exp X

• exp G

(Cohn, Livingstone, 1965)

◮ Let X ≤ V(ZG) finite,

◮ is X ∼RG H ≤ G?

Questions

How strong is the connection between the group G and the set of torsion subgroups in V(ZG)? For example, for finite X ≤ V(ZG) we have

◮ |X|

• |G|

(ˇ Zmud, Kurenno˘ ı 1967)

◮ exp X

• exp G

(Cohn, Livingstone, 1965)

◮ Let X ≤ V(ZG) finite,

◮ is X ∼RG H ≤ G? ◮ is X ≃ H ≤ G?

Questions

How strong is the connection between the group G and the set of torsion subgroups in V(ZG)? For example, for finite X ≤ V(ZG) we have

◮ |X|

• |G|

(ˇ Zmud, Kurenno˘ ı 1967)

◮ exp X

• exp G

(Cohn, Livingstone, 1965)

◮ Let X ≤ V(ZG) finite,

◮ is X ∼RG H ≤ G? ◮ is X ≃ H ≤ G?

◮ ZG ≃ ZH

= ⇒ ? G ≃ H?

Example: S3

ϕ: ZS3

∼

→ Z ⊕ Z ⊕ Z 3Z Z Z

• 2

3 3

(1, −1, −2 −3

1 2

• ) ∈ ϕ(S3)

(1, −1, 1

. . −1

• )

Zassenhaus Conjcetures

Source: S. Sehgal, Torsion units in group rings. Methods in ring theory (Antwerp, 1983)

(First) Zassenhaus Conjecture (Zassenhaus 1960s)

(ZC1) For u ∈ V(ZG) of finite order there exist x ∈ U(QG) and g ∈ G such that x−1ux = g.

(First) Zassenhaus Conjecture (Zassenhaus 1960s)

(ZC1) For u ∈ V(ZG) of finite order there exist x ∈ U(QG) and g ∈ G such that x−1ux = g.

abelian groups (Higman 1939) A5 (Luthar, Passi 1989) S5 (Luthar, Trama 1991) nilpotent groups (Weiss 1991) groups of order at most 71 (H¨

• fert 2004)

PSL(2, 7), PSL(2, 11), PSL(2, 13) (Hertweck 2004) A6 ≃ PSL(2, 9) (Hertweck 2007) metacyclic groups (Hertweck 2008) PSL(2, 8), PSL(2, 17) (Gildea; Kimmerle, Konovalov 2012) cyclic-by-abelian (Caicedo, Margolis, del R´ ıo 2013)

The prime graph (or Gruenberg-Kegel graph) of a group H is the undirected loop-free graph Γ(H) with

◮ Vertices: primes p, s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H

The prime graph (or Gruenberg-Kegel graph) of a group H is the undirected loop-free graph Γ(H) with

◮ Vertices: primes p, s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H

Prime graph question (Kimmerle 2006)

(PQ) Γ(G) = Γ(V(ZG))?

The prime graph (or Gruenberg-Kegel graph) of a group H is the undirected loop-free graph Γ(H) with

◮ Vertices: primes p, s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H

Prime graph question (Kimmerle 2006)

(PQ) Γ(G) = Γ(V(ZG))? Clearly: (ZC1) = ⇒ (PQ)

The prime graph (or Gruenberg-Kegel graph) of a group H is the undirected loop-free graph Γ(H) with

◮ Vertices: primes p, s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H

Prime graph question (Kimmerle 2006)

(PQ) Γ(G) = Γ(V(ZG))? Clearly: (ZC1) = ⇒ (PQ) solvable groups (H¨

• fert, Kimmerle 2006)

The prime graph (or Gruenberg-Kegel graph) of a group H is the undirected loop-free graph Γ(H) with

◮ Vertices: primes p, s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H

Prime graph question (Kimmerle 2006)

(PQ) Γ(G) = Γ(V(ZG))? Clearly: (ZC1) = ⇒ (PQ) solvable groups (H¨

• fert, Kimmerle 2006)

half of the sporadic simple groups (Bovdi, Konovalov, et. al. 2005 – )

The prime graph (or Gruenberg-Kegel graph) of a group H is the undirected loop-free graph Γ(H) with

◮ Vertices: primes p, s.t. there exists an element of order p in H ◮ Edges: p and q joined iff there is an element of order pq in H

Prime graph question (Kimmerle 2006)

(PQ) Γ(G) = Γ(V(ZG))? Clearly: (ZC1) = ⇒ (PQ) solvable groups (H¨

• fert, Kimmerle 2006)

half of the sporadic simple groups (Bovdi, Konovalov, et. al. 2005 – ) PSL(2, p) (Hertweck 2006)

Let x ∈ G and xG be its conjugacy class. Let u =

g∈G

ugg ∈ ZG. Then εx(u) =

• g∈xG

ug is the partial augmentation of u at (the conjugacy class of) x.

Let x ∈ G and xG be its conjugacy class. Let u =

g∈G

ugg ∈ ZG. Then εx(u) =

• g∈xG

ug is the partial augmentation of u at (the conjugacy class of) x.

Proposition (Marciniak, Ritter, Sehgal, Weiss 1987)

Let u ∈ V(ZG) be of finite order n. Then u is conjugate to an element of G in QG ⇔ εg(ud) ≥ 0 for all g ∈ G and all d | n.

Let x ∈ G and xG be its conjugacy class. Let u =

g∈G

ugg ∈ ZG. Then εx(u) =

• g∈xG

ug is the partial augmentation of u at (the conjugacy class of) x.

Proposition (Marciniak, Ritter, Sehgal, Weiss 1987)

Let u ∈ V(ZG) be of finite order n. Then u is conjugate to an element of G in QG ⇔ εg(ud) ≥ 0 for all g ∈ G and all d | n. For u ∈ ZG a normalized torsion unit of order n = 1 we have

◮ ε1(u) = 0

(Berman 1955; Higman 1939)

◮ εx(u) = 0

if o(x) ∤ n (Hertweck 2004)

Theorem (Luthar, Passi 1989; Hertweck 2004)

◮ u ∈ ZG torsion unit of order n ◮ F splitting field for G with p = char(F) ∤ n ◮ χ a (Brauer) character of F-representation D of G ◮ ζ ∈ C primitive n-th root of unity ◮ ξ ∈ F corresponding n-th root of unity

Theorem (Luthar, Passi 1989; Hertweck 2004)

◮ u ∈ ZG torsion unit of order n ◮ F splitting field for G with p = char(F) ∤ n ◮ χ a (Brauer) character of F-representation D of G ◮ ζ ∈ C primitive n-th root of unity ◮ ξ ∈ F corresponding n-th root of unity

Multiplicity µℓ(u, χ, p) of ξℓ as an eigenvalue of D(u) is given by 1 n

• d|n

TrQ(ζd)/Q(χ(ud)ζ−dℓ)

µℓ(u, χ, p) = 1 k

• d|k

TrQ(ζd)/Q(χ(ud)ζ−dℓ)

µℓ(u, χ, p) = 1 k

• d|k

TrQ(ζd)/Q(χ(ud)ζ−dℓ) = 1 k

• d|k

d=1

TrQ(ζd)/Q(χ(ud)ζ−dℓ) +1 k TrQ(ζ)/Q(χ(u)ζ−ℓ)

µℓ(u, χ, p) = 1 k

• d|k

TrQ(ζd)/Q(χ(ud)ζ−dℓ) = 1 k

• d|k

d=1

TrQ(ζd)/Q(χ(ud)ζ−dℓ)

• =:aχ,ℓ

+1 k TrQ(ζ)/Q(χ(u)ζ−ℓ)

µℓ(u, χ, p) = 1 k

• d|k

TrQ(ζd)/Q(χ(ud)ζ−dℓ) = 1 k

• d|k

d=1

TrQ(ζd)/Q(χ(ud)ζ−dℓ)

• =:aχ,ℓ

+1 k TrQ(ζ)/Q(χ(u)ζ−ℓ) As χ(u) =

C

εC(u)χ(C),

µℓ(u, χ, p) = 1 k

• d|k

TrQ(ζd)/Q(χ(ud)ζ−dℓ) = 1 k

• d|k

d=1

TrQ(ζd)/Q(χ(ud)ζ−dℓ)

• =:aχ,ℓ

+1 k TrQ(ζ)/Q(χ(u)ζ−ℓ) As χ(u) =

C

εC(u)χ(C), we obtain linear equations µℓ(u, χ, p) = tC1εC1(u) + tC2εC2(u) + ... + tChεCh(u) + aχ,ℓ. for the partial augmentations εCi(u), with “known” coefficients tCj, aχ,ℓ and µℓ(u, χ, p).

Commercial break: The HeLP-package for GAP. Using 4ti2 and soon also Normaliz.

Theorem (B., Caicedo 2015)

If G is an almost simple group with socle An, 11 ≤ n ≤ 17, then (PQ) has an affirmative answer for G.

Theorem (B., Caicedo 2015)

If G is an almost simple group with socle An, 11 ≤ n ≤ 17, then (PQ) has an affirmative answer for G.

Theorem (B., Caicedo 2015)

Let G = Sn and p, q primes bigger than n

• 3. Then there is an

element of order pq in V(ZG) if and only if there is an element of

• rder pq in G.

Some ideas of the proof

Assume u ∈ V(ZSn) with o(u) = pq and n ≥ p > n

2 ≥ q > n 3.

q.1 ↔ (•...•), q.2 ↔ (•...•)(•...•), p.1 ↔ (• • ...•)

Some ideas of the proof

Assume u ∈ V(ZSn) with o(u) = pq and n ≥ p > n

2 ≥ q > n 3.

q.1 ↔ (•...•), q.2 ↔ (•...•)(•...•), p.1 ↔ (• • ...•)

ZSn ⊆ QSn ≃ Q ⊕ Q ⊕ Mn−1(Q) ⊕ ... ⊕ Md(Q) ⊕...

λ (n) (1n) (n − 1, 1) (n − 2, 2) 1 sgn π ρ

Some ideas of the proof

Assume u ∈ V(ZSn) with o(u) = pq and n ≥ p > n

2 ≥ q > n 3.

q.1 ↔ (•...•), q.2 ↔ (•...•)(•...•), p.1 ↔ (• • ...•)

ZSn ⊆ QSn ≃ Q ⊕ Q ⊕ Mn−1(Q) ⊕ ... ⊕ Md(Q) ⊕...

λ (n) (1n) (n − 1, 1) (n − 2, 2) 1 sgn π ρ

ζ = ζpq ∈ C×: a primitive pq-th root of unity µℓ(u, χ): multiplicity of ζℓ as eigenvalue of D(u) where D ↔ χ.

Some ideas of the proof

Assume u ∈ V(ZSn) with o(u) = pq and n ≥ p > n

2 ≥ q > n 3.

q.1 ↔ (•...•), q.2 ↔ (•...•)(•...•), p.1 ↔ (• • ...•)

ZSn ⊆ QSn ≃ Q ⊕ Q ⊕ Mn−1(Q) ⊕ ... ⊕ Md(Q) ⊕...

λ (n) (1n) (n − 1, 1) (n − 2, 2) 1 sgn π ρ

ζ = ζpq ∈ C×: a primitive pq-th root of unity µℓ(u, χ): multiplicity of ζℓ as eigenvalue of D(u) where D ↔ χ.

◮ Show µ1(u, π) = 0 and µq(u, π) = 1,

Some ideas of the proof

Assume u ∈ V(ZSn) with o(u) = pq and n ≥ p > n

2 ≥ q > n 3.

q.1 ↔ (•...•), q.2 ↔ (•...•)(•...•), p.1 ↔ (• • ...•)

ZSn ⊆ QSn ≃ Q ⊕ Q ⊕ Mn−1(Q) ⊕ ... ⊕ Md(Q) ⊕...

λ (n) (1n) (n − 1, 1) (n − 2, 2) 1 sgn π ρ

ζ = ζpq ∈ C×: a primitive pq-th root of unity µℓ(u, χ): multiplicity of ζℓ as eigenvalue of D(u) where D ↔ χ.

◮ Show µ1(u, π) = 0 and µq(u, π) = 1, ◮ derive (εq.1(up), εq.2(up)) ∈ {(2, −1), (1, 0)},

Some ideas of the proof

Assume u ∈ V(ZSn) with o(u) = pq and n ≥ p > n

2 ≥ q > n 3.

q.1 ↔ (•...•), q.2 ↔ (•...•)(•...•), p.1 ↔ (• • ...•)

ZSn ⊆ QSn ≃ Q ⊕ Q ⊕ Mn−1(Q) ⊕ ... ⊕ Md(Q) ⊕...

λ (n) (1n) (n − 1, 1) (n − 2, 2) 1 sgn π ρ

ζ = ζpq ∈ C×: a primitive pq-th root of unity µℓ(u, χ): multiplicity of ζℓ as eigenvalue of D(u) where D ↔ χ.

◮ Show µ1(u, π) = 0 and µq(u, π) = 1, ◮ derive (εq.1(up), εq.2(up)) ∈ {(2, −1), (1, 0)}, ◮ derive µ1(u, ρ) ∈ {0, 1}.

System of linear equations ε(u) = 1 µ1(u, π) = 0 µ1(u, ρ) = t ∈ {0, 1} in the unknowns εq.1(u), εq.2(u), εp.1(u) and with solution

System of linear equations ε(u) = 1 µ1(u, π) = 0 µ1(u, ρ) = t ∈ {0, 1} in the unknowns εq.1(u), εq.2(u), εp.1(u) and with solution εp.1(u) t=0 = p(p − 3q) (p − q)(p − 2q) εp.1(u) t=1 = p p − 2q

System of linear equations ε(u) = 1 µ1(u, π) = 0 µ1(u, ρ) = t ∈ {0, 1} in the unknowns εq.1(u), εq.2(u), εp.1(u) and with solution εp.1(u) t=0 = p(p − 3q) (p − q)(p − 2q) εp.1(u) t=1 = p p − 2q This can not be integral except in the cases (p, q) ∈ {(3, 2), (5, 3)}

• r n = 3q − 2 and p = 2q − 1. In the latter case

(εq.1(u), εq.2(u), εp.1(u)) = (1, p, −p) is a solution to the system.

System of linear equations ε(u) = 1 µ1(u, π) = 0 µ1(u, ρ) = t ∈ {0, 1} in the unknowns εq.1(u), εq.2(u), εp.1(u) and with solution εp.1(u) t=0 = p(p − 3q) (p − q)(p − 2q) εp.1(u) t=1 = p p − 2q This can not be integral except in the cases (p, q) ∈ {(3, 2), (5, 3)}

• r n = 3q − 2 and p = 2q − 1. In the latter case

(εq.1(u), εq.2(u), εp.1(u)) = (1, p, −p) is a solution to the system. Use τ = χ(n−3,2,1).

Theorem (Kimmerle, Konovalov 2012)

Suppose that (PQ) has an affirmative answer for each almost simple image of G, then it has also a positive answer for G. [A almost simple :⇐ ⇒ ∃ S simple group, s.t. S ≤ A ≤ Aut(S).]

Theorem (Kimmerle, Konovalov 2012)

Suppose that (PQ) has an affirmative answer for each almost simple image of G, then it has also a positive answer for G. [A almost simple :⇐ ⇒ ∃ S simple group, s.t. S ≤ A ≤ Aut(S).] Simple groups having orders divisible by exactly 3 primes: A5 ≃ PSL(2, 5), PSL(2, 7), PSL(2, 8), A6 ≃ PSL(2, 9), PSL(2, 17), PSL(3, 3), PSp(3, 4) ≃ U(4, 2), U(3, 3).

Theorem (Kimmerle, Konovalov 2012)

Suppose that (PQ) has an affirmative answer for each almost simple image of G, then it has also a positive answer for G. [A almost simple :⇐ ⇒ ∃ S simple group, s.t. S ≤ A ≤ Aut(S).] Simple groups having orders divisible by exactly 3 primes: A5 ≃ PSL(2, 5), PSL(2, 7), PSL(2, 8), A6 ≃ PSL(2, 9), PSL(2, 17), PSL(3, 3), PSp(3, 4) ≃ U(4, 2), U(3, 3).

Theorem (Kimmerle, Konovalov 2012)

(PQ) has a positive answer for all almost simple groups having an

• rder divisible by 3 different primes except possibly PGL(2, 9) and

M10.

Using the “lattice method”

Theorem (B., Margolis 2013)

(PQ) has a positive answer for all almost simple groups containing

• A6. It hence has a positive answer for all groups divisible by at

most 3 different primes.

Theorem (B., Margolis 2013)

(ZC1) holds for PSL(2, 19)

More notation

◮ u ∈ V(ZG) a torsion unit ◮ p a rational prime dividing the order of u ◮ D an ordinary representation of G

More notation

◮ u ∈ V(ZG) a torsion unit ◮ p a rational prime dividing the order of u ◮ D an ordinary representation of G ◮ K the p-adic completation of a number field admitting D with

minimal ramification index over Qp

◮ R the ring of integers of K with maximal ideal P ◮ L an RG-lattice affording D ◮ k = R/P, the quotient field, and ¯ the reduction mod P

Proposition

Let o(u) = pam, p ∤ m. Let ζ ∈ R be a primitive m-th root of

• unity. Let Aj be tuples of pa-th roots of unity s.t. the eigenvalues
• f D(u) are ζA1 ∪ ζ2A2 ∪ ... ∪ ζmAm. Then, as Ru-lattice,

L ≃ M1 ⊕ ... ⊕ Mm where rankR(Mj) = |Aj| = dimk( ¯ Mj) and ¯ Mj has only one composition factor up to isomorphism.

Proposition

Let o(u) = pam, p ∤ m. Let ζ ∈ R be a primitive m-th root of

• unity. Let Aj be tuples of pa-th roots of unity s.t. the eigenvalues
• f D(u) are ζA1 ∪ ζ2A2 ∪ ... ∪ ζmAm. Then, as Ru-lattice,

L ≃ M1 ⊕ ... ⊕ Mm where rankR(Mj) = |Aj| = dimk( ¯ Mj) and ¯ Mj has only one composition factor up to isomorphism. Easiest case: K/Qp unramified, o(u) = p. Three indecomposable Ru-lattices: R, I(RCp), RCp of rank 1, p − 1, p, respectively, with corresponding eigenvalues {1}, {ξ, ..., ξp−1}, {1, ξ, ..., ξp−1}, ξ a primitive p-th root of unity. The reduction of any such lattice modulo P stays indecomposable.

Application: (ZC1) for G = PSL(2, 19)

Application: (ZC1) for G = PSL(2, 19)

After applying HeLP, only one case is left, namely: elements u ∈ V(ZG) of order 10 having partial augmentations (ε5a(u), ε5b(u), ε10a(u)) = (1, −1, 1).

Application: (ZC1) for G = PSL(2, 19)

After applying HeLP, only one case is left, namely: elements u ∈ V(ZG) of order 10 having partial augmentations (ε5a(u), ε5b(u), ε10a(u)) = (1, −1, 1). Let ζ be a 5th primitive root of unity, D18, D19 (certain) ordinary representations of G, D18 can be realized over S, a suitable extension of Z5, D19 can be realized over Z5. Let L18 and L19 be corresponding RG-lattices (note that the R’s are different), then ¯ L18 ≤ ¯ L19 and ¯ L19/¯ L18 is a trivial kG-module.

Application: (ZC1) for G = PSL(2, 19)

Using partial augmentations: D18(u) ∼ diag(

5th roots of unity

• ,

− 1, −ζ, −ζ2, −ζ3, −ζ4, −1, −ζ, −ζ4, −ζ, −ζ4) D19(u) ∼ diag(

5th roots of unity

• ,

− 1, −ζ, −ζ2, −ζ3, −ζ4, −1, −ζ, −ζ2, −ζ3, −ζ4)

Application: (ZC1) for G = PSL(2, 19)

Using partial augmentations: D18(u) ∼ diag(

5th roots of unity

• ,

− 1, −ζ, −ζ2, −ζ3, −ζ4, −1, −ζ, −ζ4, −ζ, −ζ4) D19(u) ∼ diag(

5th roots of unity

• ,

− 1, −ζ, −ζ2, −ζ3, −ζ4, −1, −ζ, −ζ2, −ζ3, −ζ4) ¯ L18 ≃ M1

18 ⊕ M−1 18 ,

¯ L19 ≃ M1

19 ⊕ M−1 19

M1

∗: trivial composition factors as k¯

u-module M−1

∗ : non-trivial composition factors as k¯

u-module

Application: (ZC1) for G = PSL(2, 19)

Using partial augmentations: D18(u) ∼ diag(

5th roots of unity

• ,

− 1, −ζ, −ζ2, −ζ3, −ζ4, −1, −ζ, −ζ4, −ζ, −ζ4) D19(u) ∼ diag(

5th roots of unity

• ,

− 1, −ζ, −ζ2, −ζ3, −ζ4, −1, −ζ, −ζ2, −ζ3, −ζ4) ¯ L18 ≃ M1

18 ⊕ M−1 18 ,

¯ L19 ≃ M1

19 ⊕ M−1 19

M1

∗: trivial composition factors as k¯

u-module M−1

∗ : non-trivial composition factors as k¯

u-module M−1

19 ∈ {2(k)− ⊕ 2I(kC5)−, (k)− ⊕ I(kC5)− ⊕ (kC5)−, 2(kC5)−}.

Application: (ZC1) for G = PSL(2, 19)

As ¯ L19/¯ L18 is a trivial kG-module, we have M−1

19 ≃ M−1 18 . But this

is impossible as the composition factors of M−1

18 as k¯

u-module cannot coincide with those of M−1

19 we just calculated.

Application: (ZC1) for G = PSL(2, 19)

As ¯ L19/¯ L18 is a trivial kG-module, we have M−1

19 ≃ M−1 18 . But this

is impossible as the composition factors of M−1

18 as k¯

u-module cannot coincide with those of M−1

19 we just calculated.

Hence, there is no such unit in question in V(ZPSL(2, 19)).

Application: (ZC1) for G = PSL(2, 19)

As ¯ L19/¯ L18 is a trivial kG-module, we have M−1

19 ≃ M−1 18 . But this

is impossible as the composition factors of M−1

18 as k¯

u-module cannot coincide with those of M−1

19 we just calculated.

Hence, there is no such unit in question in V(ZPSL(2, 19)). (ZC1) holds for PSL(2, 19)

The simple groups with four different prime divisors: PSL(2, p) A7 PSL(3, 4) PSU(3, 8) P Ω+(8, 2) PSL(2, 2f ) A8 PSL(3, 5) PSU(3, 9) Sz(8) PSL(2, 3f ) A9 PSL(3, 7) PSU(4, 3) Sz(32) A10 PSL(3, 8) PSU(5, 2) G2(3) PSL(2, 16) PSL(3, 17) PSp(4, 4)

3D4(2)

PSL(2, 25) PSL(4, 3) PSp(4, 5)

2F4(2)′

PSL(2, 27) PSU(3, 4) PSp(4, 7) M11 PSL(2, 49) PSU(3, 5) PSp(4, 9) M12 PSL(2, 81) PSU(3, 7) PSp(6, 2) J2 PSL(2, 243)

The simple groups with four different prime divisors: PSL(2, p) A7 PSL(3, 4) PSU(3, 8) P Ω+(8, 2) PSL(2, 2f ) A8 PSL(3, 5) PSU(3, 9) Sz(8) PSL(2, 3f ) A9 PSL(3, 7) PSU(4, 3) Sz(32) A10 PSL(3, 8) PSU(5, 2) G2(3) PSL(2, 16) PSL(3, 17) PSp(4, 4)

3D4(2)

PSL(2, 25) PSL(4, 3) PSp(4, 5)

2F4(2)′

PSL(2, 27) PSU(3, 4) PSp(4, 7) M11 PSL(2, 49) PSU(3, 5) PSp(4, 9) M12 PSL(2, 81) PSU(3, 7) PSp(6, 2) J2 PSL(2, 243) HeLP-method Lattice-method No method yet

Theorem (Hertweck 2006; B., Margolis 2016)

Let A be an almost simple group with socle PSL(2, pf ) for f ≤ 2. Then (PQ) has a positive answer. If S = PSL(2, p), then Aut(S) = PGL(2, p). If S = PSL(2, p2), p ≥ 5, then Out(S) = C2 × C2. PΓL(2, p2) PGL(2, p2) PΣL(2, p2) M(p2) PSL(2, p2)

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