T HE PROBABILISTIC ZETA FUNCTION OF FINITE AND PROFINITE GROUPS

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T HE PROBABILISTIC ZETA FUNCTION OF FINITE AND PROFINITE GROUPS

Andrea Lucchini

Università di Padova

Intercity number theory seminar September 3 2010, Leiden

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G a finitely generated group

a_n(G) the number of subgroups of index n in G bn(G) :=P

|G:H|=nµ_G(H)

µis the Möbius function of the subgroup lattice of G :

µ_G(H) =

( 1 if H = G

−P

H<K ≤Gµ_G(K ) otherwise

ANDREALUCCHINI THE PROBABILISTIC ZETA FUNCTION

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G a finitely generated group

an(G) the number of subgroups of index n in G b_n(G) :=P

|G:H|=nµ_G(H)

µis the Möbius function of the subgroup lattice of G :

µ_G(H) =

( 1 if H = G

−P

H<K ≤Gµ_G(K ) otherwise SUBGROUP ZETA FUNCTION

Z_G(s)=X

n∈N

a_n(G) n^s

PROBABILISTIC ZETA FUNCTION

PG(s)=X

n∈N

b_n(G) n^s

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EXAMPLE: G = Sym(3)

Sym(3) ssssssssss

JJ JJ JJ JJ J

TT TT TT TT TT TT TT TT TT

h(123)i

KK KK KK KK

KK h(12)i h(13)i

tttttttttt h(23)i jjjjjjjjjjjjjjjjjjjj h1i

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EXAMPLE: G = Sym(3)

Sym(3)1 rrrrrrrrrr

KK KK KK KK KK

UU UU UU UU UU UU UU UU UU

h(123)i

LL LL LL LL

LL h(12)i h(13)i

ssssssssss h(23)i jjjjjjjjjjjjjjjjjjjj h1i

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EXAMPLE: G = Sym(3)

Sym(3)1 ooooooooooo

NN NN NN NN NN N

WW WW WW WW WW WW WW WW WW WW WW

h(123)i − 1

OO OO OO OO OO

OO h(12)i − 1 h(13)i− 1 pppppppppppp

h(23)i − 1

gggggggggggggggggggggggg h1i

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EXAMPLE: G = Sym(3)

Sym(3)1 ooooooooooo

NN NN NN NN NN N

h(123)i − 1

OO OO OO OO OO

OO h(12)i − 1 h(13)i− 1

ppppppppppp h(23)i − 1 gggggggggggggggggggggggg

h1i3

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EXAMPLE: G = Sym(3)

Sym(3)1 ooooooooooo

NN NN NN NN NN N

h(123)i − 1

OO OO OO OO OO

OO h(12)i − 1 h(13)i− 1

h1i3

b₁(G) = µ_G(G) =1

b2(G) = µG(h(1, 2, 3)i) =−1

b3(G) = µ_G(h(1, 2)i) + µ_G(h(1, 3)i) + µ_G(h(2, 3)i) =−3 b₆(G) = µ_G(1) =3

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EXAMPLE: G = Sym(3)

Sym(3)1 ooooooooooo

NN NN NN NN NN N

h(123)i − 1

OO OO OO OO OO

OO h(12)i − 1 h(13)i− 1

h1i3

b₁(G) = µ_G(G) =1

b2(G) = µ_G(h(1, 2, 3)i) =−1

b₃(G) = µ_G(h(1, 2)i) + µ_G(h(1, 3)i) + µ_G(h(2, 3)i) =−3 b6(G) = µG(1) =3

P_G(s) =1 − 1 2^s − 3

3^s + 3

6^s Z_G(s) =1 + 1 2^s + 3

3^s + 1 6^s

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EXAMPLE: G = Z

|Z : H| = n ⇒ H = nZ µ_Z(nZ) = µ(n)

P_Z(s) =X

n∈N

µ(n)

n^s = X

n∈N

1 n^s

!−1

= (ζ(s))⁻¹= (Z_Z(s))⁻¹

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EXAMPLE: G = Z

|Z : H| = n ⇒ H = nZ µ_Z(nZ) = µ(n)

P_Z(s) =X

n∈N

µ(n)

n^s = X

n∈N

1 n^s

!−1

= (ζ(s))⁻¹= (Z_Z(s))⁻¹

EXAMPLE: G = Alt(5) P_Alt(5)(s) = 1 − 5

5^s − 6 6^s − 10

10^s + 20 20^s + 60

30^s − 60 60^s

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REFERENCES

N. Boston, A probabilistic generalization of the Riemann zeta function, Analytic number theory, Vol. 1, 155-162, Progr. Math., 138, Birkhäuser Boston, Boston, MA, 1996

A. Mann, Positively finitely generated groups, Forum Math. 8 (1996), no. 4, 429-459

K. Brown, The coset poset and probabilistic zeta function of a finite group, J. Algebra 225 (2000), no. 2, 989-1012.

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REMARK

µ_G(H) 6= 0 ⇒ H is an intersection of maximal subgroups of G.

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REMARK

The probabilistic zeta function P_G(s) depends only on the sublattice generated by the maximal subgroups of G and not on the complete subgroup lattice.

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REMARK

The probabilistic zeta function P_G(s) depends only on the sublattice generated by the maximal subgroups of G and not on the complete subgroup lattice.

The maximal subgroups of Z are the subgroups pZ, where p ranges over the set of all the prime numbers.

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REMARK

The probabilistic zeta function PG(s) depends only on the sublattice generated by the maximal subgroups of G and not on the complete subgroup lattice.

The maximal subgroups of Z are the subgroups pZ, where p ranges over the set of all the prime numbers.

The probabilistic zeta function encodes information about the lattice generated by the maximal subgroups of G, just as the Riemann zeta function encodes information about the primes.

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REMARK

If bG is the profinite completion of G, then PG(s) = P_G_b(s)

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REMARK

If bG is the profinite completion of G, then P_G(s) = P

Gb(s)

We may restrict the study of the probabilistic zeta function to the case offinitely generated profinite groups.

This allows us:

to employ properties of profinite groups;

to find a probabilistic meaning of this object.

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REMARK

This allows us:

TheFrattini subgroupFrat(G) of a profinite group G is the intersection of the (closed) maximal subgroups of G.

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REMARK

This allows us:

TheFrattini subgroupFrat(G) of a profinite group G is the intersection of the (closed) maximal subgroups of G.

µ_G(H) 6= 0 ⇒ Frat(G) ≤ H

⇓

P_G(s) = P_{G/ Frat(G)}(s).

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C ONVERGENCE AND PROBABILISTIC MEANING

QUESTION

What are the groups G for which P_G(s) converges (absolutely) in some half complex plane?

When P_G(s) converges in a suitable complex plane, which information is given by the corresponding complex function? In particular, what is the meaning of the number P_G(k ), if k is a

"large" positive integer?

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THEOREM(P. HALL1936)

If G is a finite group and t ∈ N, then PG(t) is equal to the probability that a random t-tuple generates G.

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PROOF.

It follows immediately from the Möbius inversion formula:

X

H≤G

PH(t)|H|^t = |G|^t ⇒ X

H≤G

µ_G(H)|H|^t =P_G(t)|G|^t.

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EXAMPLE: G = Alt(5)

P_G(1) = 0 since G is not a cyclic group.

PG(2) = ¹⁹₃₀ :there are 3600 = 60²ordered pairs (x , y ) ∈ G²and 2280 = ^19·3600₃₀ of these satisfy the condition hx , y i = G.

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EXAMPLE: G = Alt(5)

P_G(1) = 0 since G is not a cyclic group.

PG(2) = ¹⁹₃₀ :there are 3600 = 60²ordered pairs (x , y ) ∈ G²and 2280 = ^19·3600₃₀ of these satisfy the condition hx , y i = G.

REMARK

In 1998 Shareshian proved that if G is a finite nonabelian simple group, then P_G(s) has a multiple zero at s = 1.

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On a profinite group G, a normalized Haar measureνis defined:

Prob_G(k )= ν({(x₁, . . . ,x_k) ∈G^k | hx₁, . . . ,x_ki = G}) is the probability that k random elements generate G.

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Prob_G(k )= ν({(x₁, . . . ,x_k) ∈G^k | hx₁, . . . ,x_ki = G})

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Prob_G(k )= ν({(x₁, . . . ,x_k) ∈G^k | hx₁, . . . ,x_ki = G})

WARNING

G can be generated by k -elements 6⇒ ProbG(k ) > 0.

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Prob_G(k )= ν({(x₁, . . . ,x_k) ∈G^k | hx₁, . . . ,x_ki = G}) WARNING

G can be generated by k -elements 6⇒ Prob_G(k ) > 0.

EXEMPLE(KANTOR,LUBOTZKY1990) G =Q

n≥¯nAlt(n)^n!/8is 2-generated but Prob_G(k ) = 0 for all k ∈ N.

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Prob_G(k )= ν({(x₁, . . . ,x_k) ∈G^k | hx₁, . . . ,x_ki = G}) WARNING

n≥¯nAlt(n)^n!/8is 2-generated but Prob_G(k ) = 0 for all k ∈ N.

DEFINITION

G isPositivelyFinitelyGenerated when Prob_G(k ) > 0 for some k ∈ N.

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Prob_G(k )= ν({(x1, . . . ,xk) ∈G^k | hx₁, . . . ,xki = G}) WARNING

n≥¯nAlt(n)^n!/8is 2-generated but ProbG(k ) = 0 for all k ∈ N.

DEFINITION

G isPositivelyFinitelyGenerated when Prob_G(k ) > 0 for some k ∈ N.

THEOREM(MANN, SHALEV1996)

G is PFG ⇔ G hasPolinomialMaximalSubgroupGrowth (the number of maximal subgroups of index n is at most some given power of n).

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CONJECTUREA (MANN1996)

If G is a PFG group, then the function Prob_G(k ) can be interpolated in a natural way to an analytic function defined for all s in some right half-plane of the complex plane.

CONJECTUREB (MANN2004)

Let G be a PFG group. Then the infinite sum X

H≤oG

µ_G(H)

|G : H|^s converges absolutely in some right half plane.

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CONJECTUREA (MANN1996)

If G is a PFG group, then the function Prob_G(k ) can be interpolated in a natural way to an analytic function defined for all s in some right half-plane of the complex plane.

CONJECTUREB (MANN2004)

Let G be a PFG group. Then the infinite sum X

H≤oG

µ_G(H)

|G : H|^s converges absolutely in some right half plane.

If Conjecture B holds, then P_G(s) is absolutely convergent in a suitable half complex plane and P_G(k ) = Prob_G(s) if k ∈ N is large enough.

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Conjecture B have been proved in the following particular cases:

G is a finitely generated prosolvable group (AL 2007);

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for example, if G = bZ, then G is procyclic but PG(1) = 0, PG(2) = _ζ(2)¹ = ⁶

π² is "the probability that two integers are coprime".

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G has polynomial subgroup growth (AL 2009);

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G is an adelic group (a closed subgroup of SL(m, ˆZ)) (AL 2009);

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G is an adelic group (a closed subgroup of SL(m, ˆZ)) (AL 2009);

all the nonabelian composition factors of G are alternating groups (V. Colombo AL 2010).

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AREDUCTION THEOREM, AL 2010 Conjecture B holds if the following is true:

CONJECTURE

There exists a constant γ such that for each finite almost simple group X (S ≤ X ≤ Aut S for some finite nonabelian simple group S) we have:

|µ_X(Y )| ≤ |X : Y |^γ for each Y ≤ X .

for each n ∈ N, X contains at most n^γ subgroups Y with index n and µX(Y ) 6= 0.

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AREDUCTION THEOREM, AL 2010 Conjecture B holds if the following is true:

CONJECTURE

There exists a constant γ such that for each finite almost simple group X (S ≤ X ≤ Aut S for some finite nonabelian simple group S) we have:

|µ_X(Y )| ≤ |X : Y |^γ for each Y ≤ X .

for each n ∈ N, X contains at most n^γ subgroups Y with index n and µ_X(Y ) 6= 0.

V.COLOMBOAL 2010

The previous conjecture holds if X is an alternating or symmetric group.

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E ULER FACTORIZATION ?

If G is a finite group, then PG(s) belongs to the ringsRof the finite Dirichlet series (Dirichlet polynomials) with integer coefficients.

R is a unique factorization domain and it is interesting to study how P_G(s) factorizes in this ring.

Let N E G be a normal subgroup of G. To the factor group G/N the Dirichlet series P_G/N(s) is associated. How are the two series P_G(s) and P_G/N(s) related ?

EXEMPLE

G = Sym(3) , N = h(1, 2, 3)i.

P_G(s) = 1 − 1 2^s − 3

3^s + 3

6^s P_G/N(s) = 1 − 1 2^s P_G/N(s) divides P_G(s); indeed P_G(s) = P_G/N(s) 1 − ₃³s .

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This “good behavior” of the series P_G(s) and P_G/N(s) holds for any finite group G and any normal subgroup N:

P_G(s) = P_G/N(s)P_G,N(s)

with P_G,N(s)=Xb_n(G, N)

n^s and b_n(G, N) = X

|G:H|=n HN=G

µ_G(H)

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This “good behavior” of the series P_G(s) and P_G/N(s) holds for any finite group G and any normal subgroup N:

P_G(s) = P_G/N(s)P_G,N(s)

with PG,N(s)=Xbn(G, N)

n^s and bn(G, N) = X

|G:H|=n HN=G

µ_G(H)

If G/N can be generated by k elements, then P_G,N(k ) is the conditional probability that k elements g₁, . . . ,g_k generate G, given that hg₁, . . . ,g_kiN = G.

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Assume that1 = Nt E · · · E N0=Gis a normal series of G.

... ...

qNt =1

⇒ P_G/N_t_,N_t−1_/N_t(s)

⇒ P_G/N_i+1_,N_i_/N_i+1(s)

⇒ P_G/N₂_,N₁_/N₂(s)

qN_t−1 qN_i+1 qN_i qN₂ qN1

qN0=G

⇒ P_G/N₁_,G/N₁(s) = P_G/N₁(s)

P_G(s) =Y

i

P_G/N_i+1_,N_i_/N_i+1(s)

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CHIEF SERIES

A finitely generated profinite group G possesses a chain of open normal subgroups

G = N0D N1D . . . Ni D . . . such that

T

iN_i =1

Ni/Ni+1is a minimal normal subgroup of G/Ni+1.

For each i, the Dirichlrt seriesPi(s)=P_G/N_i+1_,N_i_/N_i+1(s) is finite and

P_G(s) =Y

i

P_i(s)

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CHIEF SERIES

T

iNi =1

N_i/N_i+1is a minimal normal subgroup of G/N_i+1.

For each i, the Dirichlrt seriesP_i(s)=P_G/N_i+1_,N_i_/N_i+1(s) is finite and

PG(s) =Y

i

Pi(s)

This infinite formal product can be computed since for each i we have Pi(s) = 1 +_m^aⁱs

i

+ . . . and for each n > 1 there exist only finitely many indices i with mi ≤ n.

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CHIEF SERIES

T

iNi =1

N_i/N_i+1is a minimal normal subgroup of G/N_i+1.

For each i, the Dirichlrt seriesP_i(s)=P_G/N_i+1_,N_i_/N_i+1(s) is finite and

PG(s) =Y

i

Pi(s)

THEOREM(E DETOMI, AL 2005)

The family of the Dirichlet polynomials {P_i(s)}_i does not depend on the choice of the chief series.

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We need some technical details to describe how the Dirichlet series Pi(s) can be computed.

We say that A = X /Y is achief factorof the profinite group G if Y ≤ X are open normal subgroups of G and X /Y is a minimal normal subgroup of G/X .

DEFINITION

Two chief factors A and B of G areG-equivalent(A ∼GB) if either A ∼=G B or there exists an open normal subgroup K of G such that G/K is a primitive permutation group (there exists a maximal

subgroup M of G containing K and K is the largest normal subgroup of G contained in M) with two different normal subgroups N₁and N₂ and A ∼=_G N₁,B ∼=_GN₂.

Two abelian chief factors are G-equivalent if and only if they are G-isomorphic, but for nonabelian chief factors G-equivalence is strictly weaker than G-isomorphism. For example the two direct factors of G = Alt(5) × Alt(5) are G-equivalent but not G-isomorphic.

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Let A = X /Y be achief factorof G and letQ(s):=P_{G/Y ,X /Y}(s).

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Let A = X /Y be achief factorof G and letQ(s):=PG/Y ,X /Y(s).

We say that A is aFrattini chief factorof G if X /Y ≤ Frat(G/Y ).

If A is a Frattini chief factor of G, then Q(s) = 1.

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Themonolithic primitive group associated with Ais defined as

L_A=

(A o (G/CG(A)) if A is abelian, G/C_G(A) otherwise.

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L_A=

If A is non-Frattini, then LAis an epimorphic image of G.

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LA=

L_Ahas a unique minimal normal subgroup, say N, and N ∼= A.

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LA=

Pe_L_A_,1(s):=P_L_A_,N(s), Pe_L_A_,i(s):=P_L_A_,N(s)−(1 + q + · · · + qⁱ⁻²)γ

|N|^s

with γ = |CAut N(LA/N)| and q =

(| End_LN| if N is abelian,

1 otherwise.

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LA=

|N|^s

1 otherwise.

Q(s) = ePLA,t(s), withtthe number of non-Frattini factors G-equivalent to A in a chief series of G/Y .

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LA=

|N|^s

1 otherwise.

A = X /Y ∼= S^r with S a finite simple group and r ∈ N;

If A is abelian, then Q(s) = 1 − c/q^swhere c is the number of complements of X /Y in G/Y .

If A is non abelian, then there exists K with S E K ≤ Aut S such that Q(s) is "approximated" by P_{K ,S}(rs −r+1).

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G = Cp× C_p

Let N ∼= Cpbe a minimal normal subgroup of G.

G/N ∼= C_p⇒ P_G/N(s) = 1 − 1/p^s.

N has complements in G ⇒ P_G,N(s) = 1 − p/p^s. Hence P(s) = (1 − 1/p^s)(1 − p/p^s).

G = Alt(5) × Alt(5)

1− 5

5^s− 6 6^s−10

10^s+20 20^s+60

30^s−60 60^s

1−5 5^s − 6

6^s−10 10^s+20

20^s+60 30^s−180

60^s

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G = Alt(5) o C2

The wreath product G = Alt(5) o C2has a unique minimal normal subgroup N ∼= Alt(5) × Alt(5).

P_G,N(s) = 1 − 25 25^s − 36

36^s −120 60^s − 100

100^s + 600

300^s + 720

360^s + 400 400^s +1200

600^s +1800

900^s − 2400

1200^s − 7200

1800^s + 3600 3600^s can be approximated by

P_Alt(5)(2s − 1) = 1 − 25 25^s − 36

36^s − 100

100^s + 400

400^s +1800

900^s − 3600 3600^s

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Let G be a finitely generated profinite group and let A be a chief factor of G. The numberδ_G(A)of non-Frattini factors in a chief series that are G-equivalent to A is finite and independent on the choice of the chief series.

PG(s) =Y

A



 Y

1≤i≤δG(A)

PeLA,i(s)





where A runs over a set of representatives of the equivalence classes of non-Frattini chief factors of G.

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LetAbe a set of representatives for the G-equivalence classes of chief factors of G.

For each simple group T , letA_T := {A ∈ A | A ∼= T^r ∃t ∈ N}.

Define

P_G^T(s)= Y

A∈AT



 Y

1≤i≤δG(A)

PeLA,i(s)





We get an Euler factorization indexed over the finite simple groups:

P_G(s) =Y

T

P_G^T(s)

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Assume that G is prosolvable:

A_T 6= ∅ ⇒ T ∼= Cp,a cyclic group of order p, for a suitable p;

for each prime p we have P_G^C^p(s) = PG,p(S) =P

m b_pm(G)

p^ms .

⇓ P_G(s) =Y

p

P_G,p(s).

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A_T 6= ∅ ⇒ T ∼= Cp,a cyclic group of order p, for a suitable p;

for each prime p we have P_G^C^p(s) = P_G,p(S) =P

m b_pm(G)

p^ms .

⇓ PG(s) =Y

p

PG,p(s).

EXAMPLE: G = Sym(4)

1 E K ∼= C₂× C₂E Alt(4) E Sym(4) P_G(s) =

1 − 1

2^s

1 − 3

3^s

1 − 4

4^s

=

1 − 1

2^s − 4 4^s + 4

8^s

1 − 3

3^s

=1 − 1 2^s − 3

3^s − 4 4^s + 3

6^s + 4 8^s + 12

12^s − 12 24^s

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A_T 6= ∅ ⇒ T ∼= C_p,a cyclic group of order p, for a suitable p;

for each prime p we have P_G^C^p(s) = P_G,p(S) =P

m b_pm(G)

p^ms .

⇓ P_G(s) =Y

p

P_G,p(s).

THEOREM(DETOMI, AL 2004) The following are equivalent:

G is prosolvable;

PG(s) =Q

pPG,p(s);

The sequence {b_n(G)}_n∈Nis multiplicative.

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A BOUT THE PROOF

Assume that P_G(s) =Q

iP_i(s) is multiplicative, with P_i(s) the Dirichlet polynomial associated to the chief factor N_i/N_i+1.

If G is finite, then Pi(s) is multiplicative for each i ∈ I. Butif Ni/Ni+1is non abelian, then Pi(s) cannot be multiplicative(this follows easily by a result proved by Guralnick: PSL(2, 7) is the unique simple group which has subgroups of two different prime power indices).

If G is infinite, then the infinite product PG(s) =Q

iPi(s) can be multiplicative even if the factors Pi(s) are not multiplicative. For this reason the proof is much more complicate.

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A PPLICATIONS - G ENERATION

QUESTION

How many random elements must be taken in a finite group G to have a “good” probability of generating G with these elements?

THEOREM(DETOMI, AL 2004)

Given a real number 0 < α < 1, there exists a constant cαsuch that for any finite group G

PG([d (G)+cα(1 + logλ(G))]) ≥ α,

whered (G)is the minimal number of generators for G andλ(G) denotes the number of non-Frattini factors in a chief series of G.

COROLLARY

If n is large enough, then [n/2] randomly chosen elements of a permutation group G of degree n almost certainly generate G.

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A PPLICATIONS - T HE COSET POSET

DEFINITION(S. BOUC- K. BROWN)

Thecoset poset C(G)associated to a finite group G is the poset consisting of the proper (right) cosets Hx , with H < G and x ∈ G, ordered by inclusion.

Hx ⊆ Ky ⇐⇒ H ≤ K and Ky = Kx .

We can apply topological concepts to this poset by using the

simplicial complex∆ = ∆(C(G))whose simplices are the finite chains H₁g₁<H₂g₂< ... <H_ng_n of elements of C(G).

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In particular, we can speak of theEuler characteristic of C(G) χ(C(G)):=X

m

(−1)^mα_m

where αmis the number of chains in C(G) of length m and we can speak of thereduced Euler characteristic of C(G)

χ(C(G))˜ := χ(C(G)) − 1.

χ(C(G)) 6= 0 ⇒ the simplicial complex ∆(C(G)) is not contractible.˜

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BOUC, BROWN2000

χ(C(G)) = −P˜ G(−1).

CONJECTURE(BROWN, 2000)

Let G be a finite group. Then P_G(−1) 6= 0, hence the simplicial complex associated to the coset poset of G is non-contractible.

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BOUC, BROWN2000

χ(C(G)) = −P˜ G(−1).

EVIDENCES FOR THE CONJECTURE

1 P_G(−1) 6= 0 for all the finite groups for which P_G(−1) have been computed by GAP.

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BOUC, BROWN2000

χ(C(G)) = −P˜ G(−1).

2 (K. Brown - 2000) True for solvable groups.

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BOUC, BROWN2000

χ(C(G)) = −P˜ _G(−1).

P_G(s) =Q

i(1 − c_i/q_i^s)with c_i ≥ 0 ⇒ P_G(−1) =Q

i(1 − c_iq_i) 6=0.

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BOUC, BROWN2000

χ(C(G)) = −P˜ G(−1).

3 (M. Patassini - 2008) True for PSL(2, q),²B2(q),²G2(q).

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BOUC, BROWN2000

χ(C(G)) = −P˜ G(−1).

3 (M. Patassini - 2008) True for PSL(2, q),²B2(q),²G2(q).

4 (M. Patassini - 2009) True for classical groups.

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P

_G

(s) WHEN G IS A FINITE GROUP

Some properties of G can be recognized from P_G(s) : DETOMI- AL G solvable ⇔ PG(s) multiplicative

(i.e. (n, m) = 1 ⇒ bnm(G) = bn(G)bm(G)).

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P

_G

(s) WHEN G IS A FINITE GROUP

Some properties of G can be recognized from P_G(s) : DETOMI- AL G solvable ⇔ P_G(s) multiplicative

(i.e. (n, m) = 1 ⇒ bnm(G) = bn(G)bm(G)).

DAMIAN- AL G p-solvable ⇔ PG(s) p-multiplicative (i.e. (p, m) = 1 ⇒ bp^rm(G) = bp^r(G)bm(G)).

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P

_G

(s) WHEN G IS A FINITE GROUP

Some properties of G can be recognized from PG(s) : DETOMI- AL G solvable ⇔ P_G(s) multiplicative

(i.e. (n, m) = 1 ⇒ bnm(G) = bn(G)bm(G)).

DAMIAN- AL G p-solvable ⇔ P_G(s) p-multiplicative (i.e. (p, m) = 1 ⇒ bp^rm(G) = bp^r(G)bm(G)).

DAMIAN- AL The set of the prime divisors of |G| coincides with the set of the primes p such that p divides m for some m with bm(G) 6= 0.

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P

_G

(s) WHEN G IS A FINITE GROUP

Some properties of G can be recognized from PG(s) : DETOMI- AL G solvable ⇔ P_G(s) multiplicative

(i.e. (n, m) = 1 ⇒ bnm(G) = bn(G)bm(G)).

DAMIAN- AL The set of the prime divisors of |G| coincides with the set of the primes p such that p divides m for some m with bm(G) 6= 0.

MASSA- AL G perfect ⇔ n divides bn(G) per each n ∈ N.

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P

_G

(s) WHEN G IS A FINITE GROUP

Some properties of G can be recognized from P_G(s) : DETOMI- AL G solvable ⇔ P_G(s) multiplicative

(i.e. (n, m) = 1 ⇒ bnm(G) = bn(G)bm(G)).

DAMIAN- AL The set of the prime divisors of |G| coincides with the set of the primes p such that p divides m for some m with b_m(G) 6= 0.

MASSA- AL G perfect ⇔ n divides bn(G) per each n ∈ N.

DAMIAN- MASSA- AL From the series P_G(s), we can deduce whether G/ Frat G is a simple group.

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THEOREM

Let G be a finite group and letm=min{n > 1 | bn(G) 6= 0}. The factor group G/ Frat G issimple and nonabelianif and only if the following conditions are satisfied:

if b_n(G) 6= 0, then n divides m!;

if q = p^r is a prime power and bq(G) 6= 0, then either bq(G) ≡ 0 mod p and q = m or (q, m) = (8, 7);

Q

n disparin^{bn (G)}ⁿ 6= 1.

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THEOREM

Let G be a finite group and letm=min{n > 1 | b_n(G) 6= 0}. The factor group G/ Frat G issimple and nonabelianif and only if the following conditions are satisfied:

if b_n(G) 6= 0, then n divides m!;

if q = p^r is a prime power and bq(G) 6= 0, then either bq(G) ≡ 0 mod p and q = m or (q, m) = (8, 7);

Q

The third condition is equivalent to say that 1 is a simple zero of the functionP

n oddbn(G)/n^s.

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THEOREM

Let G be a finite group and letm=min{n > 1 | bn(G) 6= 0}. The factor group G/ Frat G issimple and nonabelianif and only if the following conditions are satisfied:

if bn(G) 6= 0, then n divides m!;

if q = p^r is a prime power and b_q(G) 6= 0, then either b_q(G) ≡ 0 mod p and q = m or (q, m) = (8, 7);

Q

The third condition is equivalent to say that 1 is a simple zero of the functionP

n oddb_n(G)/n^s.

THEOREM(DAMIAN, PATASSINI, AL)

Let G1and G2be two finite groups with PG1(s) = PG2(s). If G1is a simple group, then G2/Frat G2∼= G1.

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I RREDUCIBILITY OF P

_G

(s)

We have seen that for any non-Frattini normal subgroup of G, the Dirichlet polynomial P_G(s) admits a corresponding non trivial factorization.

Do there exist examples of groups G such that P_G(s) has a non trivial factorization which does not come from normal subgroups?

In particular is P_G(s) irreducible in R when G is a simple group?

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I RREDUCIBILITY OF P

_G

(s)

In particular is PG(s) irreducible in R when G is a simple group?

THE SECOND QUESTION HAS A NEGATIVE ANSWER

PPSL(2,7)(s) = 1 −14 7^s − 8

8^s + 21 21^s + 28

28^s + 56 56^s − 84

84^s

=

1− 2

2^s

1+ 2

2^s+ 4 4^s−14

7^s− 28 14^s− 28

28^s+ 21 21^s+ 42

42^s

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I RREDUCIBILITY OF P

_G

(s)

In particular is PG(s) irreducible in R when G is a simple group?

THEOREM(PATASSINI2009)

Let S be a simple group of Lie type. P_S(s) is reducible if and only if S = PSL(2, p) with p = 2ⁿ− 1 a Mersenne prime and n ≡ 3 mod 4.

THEOREM(PATASSINI2010)

P_Alt(n)(s) is irreducible if n is sufficiently large.

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Even if there is not a complete correspondence between the

irreducible factors of the Dirichlet series PG(s) and the factor groups in a chief series of G, factorizing P_G(s) in the ring R can give useful information on the structure of G.

THEOREM(DAMIAN, AL 2005)

Let G be a finite solvable group. The series PG(s) can be written in a unique way in the form

PG(s) =

1 − c1

q₁^s

. . .

1 − ct

q_t^s

where q1, ..,qt are not necessarily distinct prime powers.

A chief series of G contains exactly t non-Frattini factors, with order q₁, . . .q_t.

c1+ · · · +ct is the number of maximal subgroups of G.

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Let G be a finite group and let 1 = NtE Nt−1E · · · E N0=G be a chief series of G. Define:

P_i(s) := P_N_i_/N_i+1_,G/N_i+1(s)

A:= {i | Ni/Ni+1is abelian}, B:= {i | Ni/Ni+1is not abelian}

P_G,ab(s):=Q

i∈AP_i(s), P_G,nonab(s):=Q

i∈BP_i(s) The factorization

P_G(s) = P_G,ab(s)P_G,nonab(s)

can be determined only from the knowledge of P_G(s).

PATASSINI2010 PG,ab(s) =Q

pPG,ab,p(s)

p odd =⇒ P_G,ab,p(s) = (P_G,p(s), P_G(s))

P_G,ab,2(s) = (P_G,2(s), P_G(s))/Q(s), with Q(s) ∈ R uniquely determined from the knowledge of the irreducible factors of P_G(s).

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G a finitely generated profinite group.

a_n(G) := the number of open subgroups of index n.

mn(G) := the number of maximal open subgroups of index n.

µthe Möbius function of the subgroup lattice of G :

µ_G(H) =

( 1 if H = G

−P

H<K ≤Gµ_G(K ) otherwise b_n(G) :=P

|G:H|=nµ_G(H)

A formal Dirichlet series P_G(s) can be defined by considering the generating function associated with this sequence:

P_G(s)=X

n∈N

bn(G) n^s

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Many important results have been obtained about the asymptotic behavior of the sequences {an(G)}_n∈Nand {mn(G)}_n∈N.

In particular the connection between the growth type of these sequences and the structure of G has been widely studied.

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Many important results have been obtained about the asymptotic behavior of the sequences {a_n(G)}_n∈Nand {m_n(G)}_n∈N.

It is unexplored the asymptotic behavior of the sequence {bn(G)}_n∈N. For example it would be interesting to characterize the groups G for which the growth of the sequence {bn(G)}_n∈Nis polynomial.

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It is unexplored the asymptotic behavior of the sequence {b_n(G)}_n∈N. For example it would be interesting to characterize the groups G for which the growth of the sequence {b_n(G)}_n∈Nis polynomial.

Interesting but hard question!

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It is unexplored the asymptotic behavior of the sequence {b_n(G)}_n∈N. For example it would be interesting to characterize the groups G for which the growth of the sequence {b_n(G)}_n∈Nis polynomial.

Interesting but hard question! Let us start with something (hopefully) easier.

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QUESTION

What can we say about G, if b_n(G) = 0 for almost all n ∈ N ?

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QUESTION

Before making a conjecture, let us answer to the same question for the other two sequences.

an(G) = 0 for almost all n ⇒ G is finite.

m_n(G) = 0 for almost all n ⇒ G has only finitely many maximal subgroups (which is equivalent to say that |G : Frat G| is finite).

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QUESTION

The Frattini subgroupFrat Gis the intersection of all maximal subgroups of G

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QUESTION

µ_G(H) 6= 0 ⇒ H is an intersection of maximal subgroups.

One can expect that bn(G) =P

|G:H|=nµ_G(H) = 0 for almost all n would imply that there are only finitely many subgroups of G that are intersection of maximal subgroups.