On the Prime Graph Question for 4-primary Groups I Andreas B achle - - PowerPoint PPT Presentation

On the Prime Graph Question for 4-primary Groups I

Andreas B¨ achle and Leo Margolis

Vrije Universiteit Brussel and Universit¨ at Stuttgart

Brock International Conference on Groups, Rings and Group Rings

July 28 to August 01, 2014

Notations

G finite group

Notations

G finite group R commutative ring with identity element 1

Notations

G finite group R commutative ring with identity element 1 RG group ring of G with coefficients in R

Notations

G finite group R commutative ring with identity element 1 RG group ring of G with coefficients in R ε augemtation map of RG, i.e. ε

• g∈G

rgg

• =

g∈G

rg.

Notations

G finite group R commutative ring with identity element 1 RG group ring of G with coefficients in R ε augemtation map of RG, i.e. ε

• g∈G

rgg

• =

g∈G

rg. U(RG) group of units of RG

Notations

G finite group R commutative ring with identity element 1 RG group ring of G with coefficients in R ε augemtation map of RG, i.e. ε

• g∈G

rgg

• =

g∈G

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

Notations

G finite group R commutative ring with identity element 1 RG group ring of G with coefficients in R ε augemtation map of RG, i.e. ε

• g∈G

rgg

• =

g∈G

rg. U(RG) group of units of RG V(RG) group of units of RG of augmentation 1 aka normalized units. U(RG) = U(R) · V(RG)

(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)

(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)

(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)

(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)

(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) SL(2, 5) (Dokuchaev, Juriaans, Polcino Milies, 1997)

(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) SL(2, 5) (Dokuchaev, Juriaans, Polcino Milies, 1997) groups of order at most 71 (H¨

• fert, 2004)

(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) SL(2, 5) (Dokuchaev, Juriaans, Polcino Milies, 1997) groups of order at most 71 (H¨

• fert, 2004)

PSL(2, 7), PSL(2, 11), PSL(2, 13) (Hertweck, 2004)

(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) SL(2, 5) (Dokuchaev, Juriaans, Polcino Milies, 1997) 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)

(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) SL(2, 5) (Dokuchaev, Juriaans, Polcino Milies, 1997) 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)

(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) SL(2, 5) (Dokuchaev, Juriaans, Polcino Milies, 1997) 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)

(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) SL(2, 5) (Dokuchaev, Juriaans, Polcino Milies, 1997) 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) Frobenius groups (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) Frobenius groups (Kimmerle, 2006) 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) Frobenius groups (Kimmerle, 2006) solvable groups (H¨

• fert, Kimmerle, 2006)

PSL(2, p), p a rational prime (Hertweck, 2007)

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) Frobenius groups (Kimmerle, 2006) solvable groups (H¨

• fert, Kimmerle, 2006)

PSL(2, p), p a rational prime (Hertweck, 2007) half of the sporadic simple groups (Bovdi, Konovalov, et. al. , 2005 – )

Let C ⊆ G be a conjugacy class and u =

g∈G

ugg ∈ RG.

Let C ⊆ G be a conjugacy class and u =

g∈G

ugg ∈ RG. Then εC(u) =

• g∈C

ug is called the partial augmentation of u at the conjugacy class C.

Let C ⊆ G be a conjugacy class and u =

g∈G

ugg ∈ RG. Then εC(u) =

• g∈C

ug is called the partial augmentation of u at the conjugacy class C.

Theorem (Berman, 1955; Higman, 1939)

Let u ∈ ZG a normalized torsion unit, u = 1. Then ε1(u) = 0.

Let C ⊆ G be a conjugacy class and u =

g∈G

ugg ∈ RG. Then εC(u) =

• g∈C

ug is called the partial augmentation of u at the conjugacy class C.

Theorem (Berman, 1955; Higman, 1939)

Let u ∈ ZG a normalized torsion unit, u = 1. Then ε1(u) = 0.

Theorem (Hertweck, 2004)

Let u ∈ ZG be a normalized torsion unit and C a conjugacy class

• f G. If the order of x ∈ C does not divide the order of u, then

εC(u) = 0.

Example: u ∈ V(ZA5), o(u) = 2 · 5 (Luthar, Passi, 1989)

Example: u ∈ V(ZA5), o(u) = 2 · 5 (Luthar, Passi, 1989)

1a 2a 3a 5a 5b χ 4 1 −1 −1 , Z(χ) = Z.

Example: u ∈ V(ZA5), o(u) = 2 · 5 (Luthar, Passi, 1989)

1a 2a 3a 5a 5b χ 4 1 −1 −1 , Z(χ) = Z.

◮ o(u5) = 2,

χ(u5) = ε2a(u5)χ(2a) = 0

Example: u ∈ V(ZA5), o(u) = 2 · 5 (Luthar, Passi, 1989)

1a 2a 3a 5a 5b χ 4 1 −1 −1 , Z(χ) = Z.

◮ o(u5) = 2,

χ(u5) = ε2a(u5)χ(2a) = 0 D(u5) ∼ diag(1, 1, −1, −1)

Example: u ∈ V(ZA5), o(u) = 2 · 5 (Luthar, Passi, 1989)

1a 2a 3a 5a 5b χ 4 1 −1 −1 , Z(χ) = Z.

◮ o(u5) = 2,

χ(u5) = ε2a(u5)χ(2a) = 0 D(u5) ∼ diag(1, 1, −1, −1)

◮ o(u6) = 5,

χ(u6) = ε5a(u6)χ(5a) + ε5b(u6)χ(5b) = (ε5a(u6) + ε5b(u6)) · (−1) = −1

Example: u ∈ V(ZA5), o(u) = 2 · 5 (Luthar, Passi, 1989)

1a 2a 3a 5a 5b χ 4 1 −1 −1 , Z(χ) = Z.

◮ o(u5) = 2,

χ(u5) = ε2a(u5)χ(2a) = 0 D(u5) ∼ diag(1, 1, −1, −1)

◮ o(u6) = 5,

χ(u6) = ε5a(u6)χ(5a) + ε5b(u6)χ(5b) = (ε5a(u6) + ε5b(u6)) · (−1) = −1 D(u6) ∼ diag(ζ, ζ2, ζ3, ζ4), ζ5 = 1 = ζ

Example: u ∈ V(ZA5), o(u) = 2 · 5 (Luthar, Passi, 1989)

1a 2a 3a 5a 5b χ 4 1 −1 −1 , Z(χ) = Z.

◮ o(u5) = 2,

χ(u5) = ε2a(u5)χ(2a) = 0 D(u5) ∼ diag(1, 1, −1, −1)

◮ o(u6) = 5,

χ(u6) = ε5a(u6)χ(5a) + ε5b(u6)χ(5b) = (ε5a(u6) + ε5b(u6)) · (−1) = −1 D(u6) ∼ diag(ζ, ζ2, ζ3, ζ4), ζ5 = 1 = ζ

◮ u = u5 · u6

• D(u) ∼ D(u5) · D(u6)

Example: u ∈ V(ZA5), o(u) = 2 · 5 (Luthar, Passi, 1989)

1a 2a 3a 5a 5b χ 4 1 −1 −1 , Z(χ) = Z.

◮ o(u5) = 2,

χ(u5) = ε2a(u5)χ(2a) = 0 D(u5) ∼ diag(1, 1, −1, −1)

◮ o(u6) = 5,

χ(u6) = ε5a(u6)χ(5a) + ε5b(u6)χ(5b) = (ε5a(u6) + ε5b(u6)) · (−1) = −1 D(u6) ∼ diag(ζ, ζ2, ζ3, ζ4), ζ5 = 1 = ζ

◮ u = u5 · u6

• D(u) ∼ D(u5) · D(u6) and χ(u) = ε2a(u) − 1
• .

Lemma (Marciniak, Ritter, Sehgal, Weiss, 1987; Luthar, Passi, 1989)

Let u ∈ V(ZG) be of finite order. u is conjugate to an element of G in QG ⇐ ⇒ εC(ud) ≥ 0 for all conjugacy classes C and all d | n.

Lemma (Marciniak, Ritter, Sehgal, Weiss, 1987; Luthar, Passi, 1989)

Let u ∈ V(ZG) be of finite order. u is conjugate to an element of G in QG ⇐ ⇒ εC(ud) ≥ 0 for all conjugacy classes C and all d | n.

Theorem (Luthar, Passi, 1989; Hertweck, 2004)

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

Lemma (Marciniak, Ritter, Sehgal, Weiss, 1987; Luthar, Passi, 1989)

Let u ∈ V(ZG) be of finite order. u is conjugate to an element of G in QG ⇐ ⇒ εC(ud) ≥ 0 for all conjugacy classes C and all d | n.

Theorem (Luthar, Passi, 1989; Hertweck, 2004)

◮ u ∈ ZG torsion unit of order n ◮ F splitting field for G with char(F) ∤ n ◮ χ a (Brauer) character of an 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)

This yields a system of inequalities for the partial augmentations εC(u) of u, assuming knowledge on the partial augmentations of the powers ud for divisors d of the order of u.

This yields a system of inequalities for the partial augmentations εC(u) of u, assuming knowledge on the partial augmentations of the powers ud for divisors d of the order of u. In the previous example (u ∈ V(ZA5), o(u) = 2 · 5) this yields: µ0(u, χ, 0) = −2/5 (ε5a(u) + ε5b(u)) µ1(u, χ, 0) = −1/10 (ε5a(u) + ε5b(u)) + 1/2 µ2(u, χ, 0) = 1/10 (ε5a(u) + ε5b(u)) + 1/2 µ5(u, χ, 0) = 2/5 (ε5a(u) + ε5b(u))

This yields a system of inequalities for the partial augmentations εC(u) of u, assuming knowledge on the partial augmentations of the powers ud for divisors d of the order of u. In the previous example (u ∈ V(ZA5), o(u) = 2 · 5) this yields: µ0(u, χ, 0) = −2/5 (ε5a(u) + ε5b(u)) µ1(u, χ, 0) = −1/10 (ε5a(u) + ε5b(u)) + 1/2 µ2(u, χ, 0) = 1/10 (ε5a(u) + ε5b(u)) + 1/2 µ5(u, χ, 0) = 2/5 (ε5a(u) + ε5b(u)) All this has been implemented in GAP using 4ti2 and the 4ti2-interface provided by Sebastian Gutsche.

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.

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 non-abelian simple group, s.t. S ≤ A ≤ Aut(S).

Theorem (Kimmerle, Konovalov, 2012)

(PQ) has a positive answer for all groups, whose order is divisible by at most three primes, if there are no units of order 6 in V(Z PGL(2, 9)) and in V(ZM10).

Theorem (Kimmerle, Konovalov, 2012)

(PQ) has a positive answer for all groups, whose order is divisible by at most three primes, if there are no units of order 6 in V(Z PGL(2, 9)) and in V(ZM10).

Theorem

There are no units of order 6 in V(Z PGL(2, 9)) and in V(ZM10).

Theorem (Kimmerle, Konovalov, 2012)

(PQ) has a positive answer for all groups, whose order is divisible by at most three primes, if there are no units of order 6 in V(Z PGL(2, 9)) and in V(ZM10).

Theorem

There are no units of order 6 in V(Z PGL(2, 9)) and in V(ZM10).

Corollary

The prime graph question (PQ) has an affirmative answer for all groups with order divisible by at most three different primes.

(Almost) Simple groups having an order divisible by exactly 4 primes were classified by

◮ Huppert, Lempken ◮ Bugeaud, Cao, Mignotte ◮ Kondrat’ev, Khramtsov

Ex.: G = PSU(3, 5), Out(G) ≃ S3

Ex.: G = PSU(3, 5), Out(G) ≃ S3

A = G.S3 G.3 G.23 G.22 G.21 G

Ex.: G = PSU(3, 5), Out(G) ≃ S3

Group G Γ(G)

• (u)

Characters PSU(3, 5).S3 "U3(5).S3"

2 3 5 7

2 · 7 5 · 7 PSU(3, 5).3 "U3(5).3"

2 3 5 7

PSU(3, 5).2 "U3(5).2"

2 3 5 7

3 · 5 3 · 7 PSU(3, 5) "U3(5)"

2 3 5 7

A = G.S3 G.3 G.23 G.22 G.21 G

Ex.: G = PSU(3, 5), Out(G) ≃ S3

Group G Γ(G)

• (u)

Characters PSU(3, 5).S3 "U3(5).S3"

2 3 5 7

2 · 7 5 · 7 χ2/1b, χ4/20a χ4/20a, χ10/84a PSU(3, 5).3 "U3(5).3"

2 3 5 7

PSU(3, 5).2 "U3(5).2"

2 3 5 7

3 · 5 3 · 7 χ3/20a, χ7/28a, χ9/56 χ7/28a PSU(3, 5) "U3(5)"

2 3 5 7

A = G.S3 G.3 G.23 G.22 G.21 G

Ex.: G = PSL(2, 81), Out(G) ≃ C4 × C2, o(u) = 3 · 5

Ex.: G = PSL(2, 81), Out(G) ≃ C4 × C2, o(u) = 3 · 5

A = G.(4 × 2) G.42 G.41 G.22 G.21 G.22 G.23 G

Ex.: G = PSL(2, 81), Out(G) ≃ C4 × C2, o(u) = 3 · 5

1 A = G.(4 × 2) 1 G.42 0 G.41 22 G.22 0 G.21 22 G.22 22 G.23 0 G

Ex.: G = PSL(2, 81), Out(G) ≃ C4 × C2, o(u) = 3 · 5

χ ∈ Irr(G), χ(1) = 41

◮ χ ↑G.21 splits in 2 irreducibles ◮ χ ↑G.22 stays irreducible 1 A = G.(4 × 2) 1 G.42 0 G.41 22 G.22 0 G.21 22 G.22 22 G.23 0 G

Ex.: G = PSL(2, 81).42, o(u) = 2 · 41

A character χ is called p-constant if it takes the same value on all conjugacy classes of elements of order p.

Ex.: G = PSL(2, 81).42, o(u) = 2 · 41

A character χ is called p-constant if it takes the same value on all conjugacy classes of elements of order p. One does not need to know the possible solutions for elements of

• rder p to obtain constraints for elements of order p · q. G has 22

irreducible 41-constant characters giving as only possibility for elements of order 2 · 41 (ε(41)(u), ε2a(u), ε2b(u)) = (0, 1, 0). But then ε(2)(u) ≡ 0 mod 2.

Lack of certain character tables

◮ Let G = PSU(3, 8), then Out(G) ≃ S3 × C3. For

A = Aut(G) = G.(S3 × C3) and G.(C3 × C3) there are no character tables available in GAP, but inducing from G gives enough information to rule out all orders in question.

Lack of certain character tables

◮ Let G = PSU(3, 8), then Out(G) ≃ S3 × C3. For

A = Aut(G) = G.(S3 × C3) and G.(C3 × C3) there are no character tables available in GAP, but inducing from G gives enough information to rule out all orders in question.

◮ There is no character table for G = PSL(3, 17) and its

automorphism group available in GAP, but a paper of Simpson and Frame provides all information needed to deal with all possible orders, except for possibly 3 · 17.

Example: G = PSp(4, 7), o(u) = 2 · 5

For G the possibility (ε2a(u), ε2b(u), ε5a(u)) ∈ {(0, 6, −5), (0, −4, 5)} can not be ruled out, solely looking at the ordinary character table

• f G. But the exceptional isomorphism PSp(4, 7) → PΩ(5, 7)

provides a Brauer character which eliminates these possibilities.

Results

◮ There are 121 specific almost simple groups having an order

divisible by exactly 4 primes and 5 (possibly infinite) series of almost simple groups having an order divisible by exactly 4 primes.

Results

◮ There are 121 specific almost simple groups having an order

divisible by exactly 4 primes and 5 (possibly infinite) series of almost simple groups having an order divisible by exactly 4 primes.

◮ 9 groups have already been treated by Bovdi, Kimmerle,

Konovalov, and Salim.

Results

◮ There are 121 specific almost simple groups having an order

divisible by exactly 4 primes and 5 (possibly infinite) series of almost simple groups having an order divisible by exactly 4 primes.

◮ 9 groups have already been treated by Bovdi, Kimmerle,

Konovalov, and Salim.

◮ 98 groups were handled by us, using HeLP and extensions.

Results

◮ There are 121 specific almost simple groups having an order

divisible by exactly 4 primes and 5 (possibly infinite) series of almost simple groups having an order divisible by exactly 4 primes.

◮ 9 groups have already been treated by Bovdi, Kimmerle,

Konovalov, and Salim.

◮ 98 groups were handled by us, using HeLP and extensions. ◮ 2 series have been treated by the HeLP-method.

To be continued...