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ISSN (print): 2251-7650, ISSN (on-line): 2251-7669 Vol. 4 No. 2 (2015), pp. 49-67.
c
⃝2015 University of Isfahan
www.ui.ac.ir
BIAS OF GROUP GENERATORS IN FINITE AND PROFINITE GROUPS: KNOWN RESULTS AND OPEN PROBLEMS
ELEONORA CRESTANI AND ANDREA LUCCHINI∗
Communicated by Patrizia Longobardi
Abstract. We analyze some properties of the distributionQG,k of the first component in ak-tuple
chosen uniformly in the set of all thek-tuples generating a finite groupG(the limiting distribution of the product replacement algorithm). In particular, we concentrate our attention on the study of the variation distanceβk(G) betweenQG,kand the uniform distribution. We review some known results,
analyze several examples and propose some intriguing open questions.
1. The product replacement algorithm and the bias of its limiting distribution
The product replacement algorithm (PRA) is a practical algorithm to construct random elements of a finite group. The algorithm was introduced and analyzed in [2], where the authors proved that it produces asymptotically uniformly distributed elements. Since then the algorithm has been widely investigated (see for example [17], [12]). The PRA is defined as follows. LetGbe a finite group and let d(G) be the minimal number of generators ofG. For any integert≥d(G),let ΦG(t) ={(g1, . . . , gt)∈
Gt | ⟨g
1, . . . , gt⟩ =G} be the set of all generating t-tuples of G. Given a generating t-tuple, a move
to another such tuple is defined by first uniformly selecting a pair (i, j) with 1≤i̸=j ≤t and then applying one of the following operations with equal probability:
R±i,j : (g1, . . . , gi, . . . , gt)7→(g1, . . . , gi·gj±1, . . . , gt),
L±i,j : (g1, . . . , gi, . . . , gt)7→(g1, . . . , g±j1·gi, . . . , gt).
MSC(2010): Primary: 20P05; Secondary: 20F69.
Keywords: Product replecement algorithm; profinite groups; group generators. Received: 11 October 2014, Accepted: 19 June 2015.
∗Corresponding author.
To produce a random element inG, start with some generatingt-tuple, apply the above moves several times, and finally return a random element of the generating t-tuple that was reached. The moves in the PRA can be conveniently encoded by the PRA graph Γt(G) whose vertices are the tuples in
ΦG(t), with edges corresponding to the moves R±i,j, L±i,j. The algorithm consists of running a nearest
neighbor random walk on this graph (a product replacement random walk) and returning a random component. Several results ensure that the graph Γk(G) is connected ifk is large enough and that a
sufficiently long product replacement random walk reaches an almost uniform distributed generating t-tuple (these results are surveyed in [17]). But even if the graph Γk(G) is connected and the product
replacement random walk mixes rapidly, the resulting distribution of the output can still be biased.
The limiting distribution of the product replacement algorithm is the probability distributionQG,t
on Gof the first components of t-tuples chosen uniformly from ΦG(t). For the product replacement
algorithm to generate “random” group elements, it is necessary thatQG,t be close toUG,the uniform
distribution on G. We estimate the bias of the distribution QG,t considering the variation distance
betweenQG,t and the uniform distributionUG:
βt(G) =∥QG,t−UG∥tv= max
B⊆G|QG,t(B)−UG(B)|=
1 2
∑
g∈G
QG,t(g)−
1
|G| .
We have 0 ≤ βt(G) ≤1 and the smaller βt(G) is, the closer is QG,t to the uniform distribution UG.
Let us give an example, considering the case when G= Sym(3) and t= 2.Since
ΦG(2) ={((1,2),(1,2,3)), ((1,3),(1,2,3)), ((2,3),(1,2,3)),((1,2,3),(1,2)),
((1,2,3),(1,3)), ((1,2,3),(2,3)), ((1,2),(1,3,2)), ((1,3),(1,3,2)), ((2,3),(1,3,2)), ((1,3,2),(1,2)), ((1,3,2),(1,3)), ((1,3,2),(2,3)), ((1,2),(1,3)), ((1,2),(2,3)), ((1,3),(2,3)), ((1,3),(1,2)),
((2,3),(1,2)), ((2,3),(1,3))}
we have
QG,2(g) =
0 ifg= 1 4
18 ifg= (i, j) 3
18 ifg= (i, j, k) and therefore
β2(G) = 1 2
∑
g∈G
QG,2(g)−
1
|G| =
1 2
( 0−
1 6
+ 3
4 18 −
1 6
+ 2
3 18 −
1 6
)
= 1 6.
As we will see in Section 4 we may extend the definition of βt(G) in the context of profinite groups.
Indeed let G be a t-generated profinite group: G is the inverse limit of its finite epimorphic images G/N, where N runs in the set N of the open normal subgroups of Gand for every choice of N ∈ N
two probability distributions QG/N,t and UG/N are defined on the quotient group G/N: this allows
different ways. One of the two measures obtained in this way is the usual normalized Haar measure µG. The other measureκG,t has the property thatκG,t(X) = infN∈NQG/N,t(XN/N) for every closed
subsetX ofG. We estimate the bias of the measureκG,t considering
βt(G) =∥κG,t−µG∥tv= sup
B∈B(G)
|κG,t(B)−µG(B)|= sup N∈N
βt(G/N)
where B(G) is the set of the measurable subsets of G.
Partial results concerning the behavior of the bias βt(G) have been obtained in relation with the
analysis of the efficiency of the product replacement algorithm. We think that it should be interesting to start a more systematic study, with the aim of understanding how some generation properties of a profinite group G are encoded by the behavior of the function t→βt(G). In this paper we review
some known results, analyze several examples and propose some intriguing open questions.
2. Computing βt(G) when G is a finite group
It this section we consider the case when G is a finite group and we describe how βt(G) can be
computed.
For any positive integert,letϕG(X, t) denote the cardinality of the set ΦG(X, t) of orderedt-tuples
(g1, . . . , gt) of group elements such thatG=⟨X, g1, . . . , gt⟩.The number
PG(X, t) =
ϕG(X, t)
|G|t
is the probability thattrandomly chosen elements generateGtogether with the elements of the subset X. We will write PG(g, t) instead of PG({g}, t) andPG(t) instead of PG(∅, t).
Now let t be a positive integer withd(G) ≤t. Let QG,t be the probability distribution of the first
component of (g1, . . . , gt),where (g1, . . . , gt) is selected uniformly at random from all thet-tuples which
generate G. So ifX ⊆G, thenQG,t(X) is the probability thatg1 ∈X given that⟨g1, . . . , gt⟩=G. In
particular
QG,t(X) =
∑
x∈X|ΦG(x, t−1)|
|ΦG(t)|
=
∑
x∈XPG(x, t−1)
PG(t)|G|
.
For every g let us define
σG,t(g) =
PG(g, t−1)
PG(t)
.
Moreover let
We have
βt(G) =
1 2
∑
g∈G
QG,t(g)−
1
|G| =
1 2|G|
∑
g∈G
|σG,t(g)−1|
= 1 2|G|
∑
g∈∆+ G(t)
(σG,t(g)−1) +
∑
g∈∆− G(t)
(1−σG,t(g))
.
On the other hand, ΦG(t) is the disjoint union of the subsets ΦG(g, t−1), g∈G,hence∑g∈GσG,t(g) =
|G|and therefore
∑
g∈∆+ G(t)
(σG,t(g)−1)
+
∑
g∈∆− G(t)
(σG,t(g)−1)
= 0
and
(2.1) βt(G) =
1
|G|
∑
g∈∆+ G(t)
(σG,t(g)−1)
=
1
|G|
∑
g∈∆− G(t)
(1−σG,t(g))
.
As discovered by P. Hall [7] the function PG(X, t) can be represented by a suitable Dirichlet
poly-nomial; indeed we have
(2.2) PG(X, t) =
∑
X⊆H≤G
µG(H)
|G:H|t.
where µis the M¨obius function associated with the subgroup lattice ofG. Moreover, let Nl= 1≤ · · · ≤N0=G
be a chief series of G. In [10] it is proved that to each chief factorNi−1/Ni it is associated a Dirichlet
polynomial Pi(X, t) with integer coefficients with the property that
PG(X, t) =
∏
1≤i≤l
Pi(X, t).
In the particular case when Gis soluble,
(2.3) PG(g, t) =
∏
1≤i≤l
(
1− ci(g) |Ni−1/Ni|t
)
where ci(g) is the number of complements of Ni−1/Ni inG/Ni containinggNi.
It follows from (2.2) that for any g∈G andt≥d(G) we have
σG,t(g) =
∑
g∈H≤G
µG(H)|G:H| |G:H|t ∑
H≤G µG(H) |G:H|t
.
In particular σG,t(g) → 1 as t → ∞ and 1−σG,t(g) is a monotone function if t is large enough.
Together with (2.1) this implies:
Proposition 1. βt(G) is a monotonically decreasing function when t is large enough and βt(G)→0
However the behavior of σG,t(g) can be quite unpredictable when t is small: consider for example
a cyclic groupG of order 15 and letg be an element of order 3: it follows from (2.3) that
σG,t(g) =
3t(5t−5) (3t−1)(5t−1).
The values ofσG,t(g) when t≤5 are described in the following table: Table 1.
t 1 2 3 4 5
σG,t(g) 0 15/16 405/403 837/832 94770/94501
The first surprise is that σG,2(g) < 1 while σG,3(g) > 1, or equivalently QG,2(g) < 1/15 while QG,3(g) > 1/15: the occurrence of g in a generating pair is below average but the occurrence in a generating triple is above average. Moreover σG,3(g)< σG,4(g) butσG,4(g) > σG,5(g).This behavior
of σG,t(g) for small values oftmakes the following question difficult to be approached:
Question 1. Is it true that βt1(G)≤βt2(G) whenever t1 ≥t2?
3. Normal subgroups
As one can expect, βt(G1) ≤βt(G2) wheneverG1 is an epimorphic image of G2. For the reader’s convenience, we include in this section the proof of this fact, which has been already given in [4].
Assume that N is a normal subgroup of the factor group G. First we need to study the relation between the two probability distributionsQG,t andQG/N,t.LetG=G/N and, for anyg∈G, denote
by g the elementgN of G.
Lemma 2. QG,t(gN) =QG,t(g).
Proof. We may assumeQG,t(g)̸= 0,otherwise we would have QG,t(gN) = QG,t(g) = 0.In particular
there exist x1, . . . , xt−1 ∈ G such that G =⟨g, x1, . . . , xt−1, N⟩.If t∈ N, X ⊆G and PG(X, t) ̸= 0,
thenPG,N(X, t) =PG(X, t)/PG(X, t) expresses the conditional probability thattrandom elements of
GgenerateGtogether with the elements ofX given that they generateGtogether with those ofXN. In our casePG,N(t)|N|t is the cardinality of the set
Ω ={(n, n1, . . . , nt−1)∈Nt| ⟨gn, x1n1, . . . , xt−1nt−1⟩=G}, while, for anyn inN, PG,N(gn, t−1)|N|t−1 is the cardinality of the set
Ωn={(n1, . . . , nt−1)∈Nt−1 | ⟨gn, x1n1, . . . , xt−1nt−1⟩=G}. Clearly Ω is the disjoint union of the subsets {n} ×Ωn, n∈N, and therefore
∑
n∈N
Notice that
QG,t(gN)
QG,t(g) =
∑
n∈NPG(gn, t−1)
PG(t)|G|
PG(t)|G|
PG(g, t−1)|N|
=
∑
n∈NPG,N(gn, t−1)
PG,N(t)|N|
= 1
and the conclusion follows. □
Proposition 3. If N ⊴ G and t ≥ d(G), then βt(G) ≥ βt(G/N). The equality holds if and only if
(σG,t(g1)−1)(σG,t(g2)−1)≥0 whenever g1 and g2 are in the same coset of N ∈G. Proof. Letg1, . . . , gm be a transversal ofN inG. We have
βt(G) =
1 2
∑
g∈G
QG,t(g)−
1
|G| = 1 2 ∑
1≤i≤m
( ∑
n∈N
QG,t(gin)−
1
|G| ) ≥ 1 2 ∑
1≤i≤m
∑
n∈N
(
QG,t(gin)−
1
|G| ) = 1 2 ∑
1≤i≤m
QG,t(giN)−|
N| |G| = 1 2 ∑
1≤i≤m
QG,t(gi)−
1
|G|
=βt(G).
The equality holds if and only if for each i∈ {1, . . . , m} we have
∑
n∈N
QG,t(gin)− 1
|G| = ∑
n∈N
(
QG,t(gin)− 1
|G| ) or equivalently ∑
n∈N
|σG,t(gin)−1|=
∑
n∈N
(σG,t(gin)−1)
.
This is equivalent to have
(σG,t(gin1)−1)(σG,t(gin2)−1)≥0
for every n1, n2∈N. □
Clearly, if f ∈Frat(G),the Frattini subgroup of G, thenPG(g, t) =PG(gf, t) for eachg∈G. This
implies that σG,t(g1) = σG,t(g2) whenever g1Frat(G) = g2Frat(G) and therefore it follows from the previous proposition that:
Corollary 4. If N ≤FratG thenβt(G) =βt(G/N).
However the conditionβt(G) =βt(G/N) for every t≥d(G) does not implies thatN is contained in
the Frattini subgroup ofG. For example we will see in Section 9 thatβt(Sym(3)) =βt(Sym(3)/Alt(3))
4. Profinite groups
Let G be a t-generated profinite group and let N be the set of the open normal subgroups of G. The group Gis the inverse limit of the factor groupsG/N where N runs in N; for each N ∈ N, the group G/N is a finite probability space with respect to the distribution QG/N,t. If N1, N2 are open normal subgroups of GwithN1≤N2 then there is a natural epimorphism
πN1,N2 :G/N1→G/N2
and Lemma 2 ensures that if X is a subset of G containingN2,then QG/N2,t(X/N2) =QG/N1,t(π
−1
N1,N2(X/N2)) =QG/N1,t(X/N1).
In particular
({G/N, QG/N,t}N∈N,{πN,M}N,M∈N,N≤M)
is an inverse system of probability spaces (see [18, Definition 2.1]). In [18] the author gives a condition on a inverse system of measure spaces that guarantees the existence and uniqueness of a limit measure space. This condition is trivially satisfied by the inverse limits of discrete probability spaces: hence there is a uniquely defined measureκG,t onGwith the property that, for anyN ∈ N and any unionX
of cosets ofN we haveκG,t(X) =QG/N,t(X/N).We estimate the bias of the measureκG,t considering
βt(G) =∥κG,t−µG∥tv= sup
B∈B(G)
|κG,t(B)−µG(B)|
where B(G) is the set of the measurable subsets of G.
Proposition 5. If G is a t-generated profinite group, then
βt(G) = sup N∈N
βt(G/N).
Proof. LetN ∈ N and letBN(G) be the set of the subsets ofGthat are union of cosets ofN inG.We
have BN(G) ⊆ B(G) and if B ∈ BN(G), then κG,t(B) = QG/N,t(B/N) and µG(B) = UG/N(B/N) =
|B/N|/|G/N|.But then
βt(G/N) = max
B∈BN(G)|QG/N,t(B/N)−UG/N(B/N)| = max
B∈BN(G)|κG,t(B)−µG(B)|
≤ sup
B∈B(G)
|κG,t(B)−µG(B)|=βt(G).
This implies
βt(G)≥ sup N∈N
βt(G/N).
To conclude our proof, it suffices to show that for every positive real number ϵ, there exists N ∈ N
such that βt(G)≤βt(G/N) +ϵ. By definition there existsB ∈ B such that
(4.1) |κG,t(B)−µG(B)| ≥βt(G)− ϵ
Notice that being G finitely generated, there exists a descending chain {Ni}i∈N of open normal sub-groups of Gsuch that∩i∈NNi= 1.For everyn∈N,let
γn(B) = |
BNn/Nn|
|G/Nn|
and δn(B) =QG/Nn,t(BNn/Nn).
We have that {γn}n∈N and {δn}n∈N are decreasing functions and µG(B) = inf
n∈Nγn(B) and κG,t(B) = infn∈Nδn(B). In particular there existsn∈Nsuch that
γn(B)−
ϵ
2 ≤µG(B)≤γn(B), δn(B)− ϵ
2 ≤κG,t≤δn(B) and consequently
(4.2) |κG,t(B)−µG(B)| ≤ |γn(B)−δn(B)|+
ϵ
2 ≤βt(G/Nn) + ϵ 2.
From (4.1) and (4.2) we immediately conclude thatβt(G)≤βt(G/Nn) +ϵ. □
5. Nilpotent and pronilpotent groups
We can now apply the results described in Sections 2 and 3 in order to compute βt(G) when G is
a t-generated pronilpotent group.
First assume thatGis a finite nilpotent group. LetI ={p1, . . . , pt}be a set of prime numbers, and
let G=∏
i∈IPi wherePi∼=Cpdii is an elementary abelianpi group of rank di>0.By (2.3) we have
PG(t) =
∏
i∈I
∏
0≤u≤di−1
(
1−p
u i pt i ) .
For any subset J ofI, let ΩJ ={g∈G| |g|=∏j∈Jpj}.Ifg∈ΩJ then by (2.3)
PG(g, t−1) =
∏
i∈J
∏
0≤u≤di−2
(
1− p
u i
pti−1
) ∏
i /∈J
∏
0≤u≤di−1
(
1− p
u i
pti−1
) . It follows: (5.1)
σG,t(g) =
PG(g, t−1)
PG(t)
=
( ∏
i∈J
1 1−p1t
i ) ∏
i /∈J
1−pdii
pt i
1−p1t i = (∏
i∈Jpti
)(∏
i /∈J(pti−pdii)
)
∏
i∈I(pti−1)
.
Hence
2βt(G)|G|=
∑
J⊆I
(∏
i∈Jpti
)(∏
i /∈J(pti−pdii)
)
∏
i∈I(pti−1)
−1
|ΩJ|
and
βt(G) =
1 2
∑
J⊆I
(∏
i∈Jpti
)(∏
i /∈J(pti−pdii)
)
∏
i∈I(pti−1)
−1 ∏
i∈J(pdii−1)
∏
i∈Ipdii
.
In particular if|G|=Cd
p andt≥d, then
PG(g, t−1)
PG(t)
=
pt−pd
pt−1 <1 ifg= 1
pt
pt−1 >1 ifg̸= 1
and by (2.1)
βt(G) =
(
1− (
pt−pd
pt−1
))
1 pd =
(
pd−1
pt−1
)
1 pd.
Since βt(G) =βt(G/FratG) for everyt-generated profinite group G,we deduce:
Proposition 6. If G is a finitely generated pro-p group and t≥d(G),then
βt(G) =
(
pd(G)−1 pt−1
)
1 pd(G).
Proposition 7. Let G be a t-generated pronilpotent group. If t≥2,then βt(G)≤
π2
6 −1∼0.645.
Proof. By Proposition 5, it suffices to prove the statement for finite nilpotent groups. Moreover, since βt(G) =βt(G/Frat(G)),we may assume thatGis a finite abelian group with a square-free exponent.
By (5.1), for every g∈G we have
σG,t(g)≤
∏
p||G| (
1− 1
pt )−1 ≤ ∞ ∏ p (
1− 1
p2
)−1
=ζ(2) = π 2 6 . But then, by (2.1), we have
βt(G) =
1
|G|
∑
g∈∆+G(t)
(σG,t(g)−1)
≤
|∆+G(t)| |G|
(
π2 6 −1
) ≤ π
2 6 −1,
hence our claim is proved. □
It is interesting to analyze in more details the caseG=⟨x⟩cyclic of orderp·q wherepandq primes with p < q.We have
σG,t(g, t) =
(pt−p)(qt−q)
(pt−1)(qt−1) <1 if|g|= 1,
σG,t(g, t) =
pt(qt−q)
(pt−1)(qt−1) if|g|=p,
σG,t(g, t) =
(pt−p)qt
(pt−1)(qt−1) <1 if|g|=q,
σG,t(g, t) = p tqt
Notice that if |g| = p, then σG,t(g, t) ≥ 1 if and only if (qt−1)/(q −1) ≥ pt, which is true if t is
large enough. If (qt−1)/(q−1)≥pt, thenσ
G,t(g, t)≤1 if and only ifg is contained in the subgroup
N =⟨xp⟩ ofG of orderq, so, by Proposition 3, β
t(G) =βt(G/N) =βt(Cp); hence
βt(Cpq) =βt(Cp) =
p−1 p(pt−1) if
qt−1 q−1 ≥p
t.
For example, if p= 2, then (qt−1)/(q−1)≥2t whenever q is an odd prime and t≥2 so βt(C2q) =
βt(C2) = (2(2t−1))−1 ift≥2.
If (qt−1)/(q−1)< pt,thenσG,t(g, t)>1 if and only if|g|=pq, so
βt(Cpq) =
ϕ(pq) pq
(
ptqt
(pt−1)(qt−1)−1
)
=(p
t+qt−1)(p−1)(q−1)
(pt−1)(qt−1)pq if
qt−1
q−1 < p
t.
In particular, for t= 1 we have
β1(Cpq) =
p+q−1 pq = 1−
ϕ(pq) pq .
This is a particular case of a more general situation: indeed if G= Cn then σG,1(g) = 0 if ⟨g⟩ ̸= G, σG,1(g) = 1/PG(1) =n/ϕ(n) otherwise, so
β1(G) = ϕ(n)
n
(
n ϕ(n) −1
)
= 1− ϕ(n)
n .
6. Comparing βt(G) and PG(t)
We concluded the previous section, showing thatβ1(G) = 1−PG(1) ifG is a cyclic group. This is
a particular instance of a more general result (see also [17, Proposition 1.5.1]):
Proposition 8. If Gis a finite group and t≥d(G),thenβt(G)≤1−PG(t). Moreover ifG̸= 1 then
Proof. Setδ(g1, . . . , gt) = 1 if⟨g1, . . . , gt⟩=G, δ(g1, . . . , gt) = 0 otherwise. We have
2βt(G) =
∑
g1∈G
QG(g1)− 1
|G|
= ∑
g1∈G
|ΦG(g1, t−1)|
|ΦG(t)| −
1
|G|
= ∑
g1∈G ∑
(g2,...,gt)∈Gt−1
δ(g1, g2, . . . , gt)
|ΦG(t)|
−
1
|G| = ∑
g1∈G ∑
(g2,...,gt)∈Gt−1 (
δ(g1, g2, . . . , gt)
|ΦG(t)| −
1
|G|t
) ≤ ∑
(g1,g2,...,gt)∈Gt
δ(g1, g2, . . . , gt)
|ΦG(t)|
− 1 |G|t
= ∑
(g1,...,gt)∈φG(t)
δ(g1, . . . , gt)
|ΦG(t)| −
1
|G|t
+ ∑
(g1,...,gt)∈/φG(t)
δ(g1, . . . , gt)
|ΦG(t)| −
1
|G|t
= ∑
(g1,...,gt)∈φG(t)
1
|ΦG(t)|−
1
|G|t
+ ∑
(g1,...,gt)∈/φG(t)
1
|G|t
=|ΦG(t)|
(
1
|ΦG(t)|
− 1 |G|t
)
+|G|
t− |Φ G(t)|
|G|t = 2
(
1−|ΦG(t)| |G|t
)
.
The equality is satisfied if and only if for each g1 ∈Gwe have that
∑
(g2,...,gt)∈Gt−1 (
δ(g1, g2, . . . , gt)
|ΦG(t)| −
1
|G|t
) = ∑
(g2,...,gt)∈Gt
δ(g1, g2, . . . , gt)
|ΦG(t)| −
1
|G|t
or equivalently if and only if
(
δ(g1, x2, . . . , xt)
|ΦG(t)| −
1
|G|t
) (
δ(g1, y2, . . . , yt)
|ΦG(t)| −
1
|G|t
)
≥0 (∗)
for each g1, x2, . . . , xt, y2, . . . , yt ∈ G. This is true if G = 1 or if G is cyclic and t = 1. Conversely
assume that (∗) is satisfied: in particular we can choose g1, x2, . . . , xt such that ⟨g1, x2, . . . , xt⟩ =G
and take y2=· · ·=yt= 1 : we have
(
1
|ΦG(t)|
− 1 |G|t
) (
δ(g1,1, . . . ,1)
|ΦG(t)|
− 1 |G|t
) ≥0,
hence δ(g1,1, . . . ,1) = 1, i.e. G=⟨g1⟩. Moreovert= 1, otherwise
(
δ(1,1, . . . ,1)
|ΦG(t)| −
1
|G|t
) (
δ(1, g1,1, . . . ,1)
|ΦG(t)| −
1
|G|t
)
= −1
|G|t
(
1
|ΦG(t)|−
1
|G|t
)
<0.
This concludes the proof of our statement. □
It is well-known that a profinite group G, being a compact topological group, can be seen as a probability space. If we denote with µ the normalized Haar measure on G, so that µ(G) = 1, the probability thatk random elements generateG is defined as
where µdenotes also the product measure onGk. By [14, Theorem 1] we have
(6.1) PG(t) = inf
N∈NPG/N(t).
But then
βt(G) = sup N∈N
βt(G/N)≤ sup N∈N
(1−PG/N(t))
= 1− inf
N∈NPG/N(t) = 1−PG(t).
So we have:
Corollary 9. If Gis a t-generated profinite group, then βt(G)≤1−PG(t).
As a consequence of Proposition 8, we are ensured that the distributionQG,t is “almost” uniform if
trandomly chosen elements from Galmost certainly generateG: in this casePG(t) is closed to 1 and
soβt(G) is closed to 0.However this condition is quite far from being necessary. Indeed the inequality
βt(G)≤1−PG(t) is not sharp. For example we have:
Theorem 10 ([4, Theorem 3]). For every positive real number ε, there exist a positive integer t and a t-generated prosupersoluble group G such thatPG(t) = 0 and βt(G)≤ε.
7. A defect in the product replacement algorithm
In [1] Babai and Pak demonstrated a defect in the product replacement algorithm. For certain groups Gthe distribution QG,t is far fromUG.More precisely:
Theorem 11 ([1, Theorem 2.1]). Let G= Alt(n)n!/8. If n≥5, then G is 2-generated but, fort≥4, βt(G) tends to 1 as n→ ∞.
The result of Babai and Pak implies that if ˆF2 is the free profinite group of rank 2 andt≥4,then βt( ˆF2) = 1. In [17] Pak proposed the following problem: can one exhibit the bias for a sequence of finite soluble groups? In other words can we produce a sequence of t-generated finite soluble groups such thatβt(Hn)→1 asn→ ∞? Equivalently does there exist at-generated prosoluble groupGwith
βt(G) = 1? It is not difficult to give an affirmative answer in the particular case when t=d(G).For
example in [3] the following result has been proved:
Theorem 12 ([3, Theorem 1]). There exists a 2-generated metabelian profinite G group with the property that µG(Ω) = 0,where
Ω ={x∈G| ⟨x, y⟩=G for some y∈G}. By definition κG,2(Ω) = 1and consequently β2(G)≥ |κG,2(Ω)−µG(Ω)|= 1.
A more important and intriguing question is whether we can find a finitely generated prosoluble group G with the property thatβt(G) = 1 for some given integer tsignificantly larger than d(G). It
follows from Corollary 9 that we can haveβt(G) = 1 only ifPG(t) = 0.If we consider arbitrary profinite
rank d≥2 then PFˆd(t) = 0 for everyt≥d(see for example [8]). The situation is different in the case of finitely generated prosoluble groups. If G is a finitely generated prosoluble group thenPG(t) >0
if t ≥ ⌈c(d(G)−1) + 1⌉,with c ⋍3.243,the P`alfy-Wolf constant (see [11] and [13]): so if G is a t-generated prosoluble group withβG(t) = 1,then the ratio betweentand the smallest cardinalityd(G)
of a generating set of G cannot be arbitrarily large. However in [3] examples have been constructed of prosoluble d-generated groups Gd for which there exists an integer td with βtd(Gd) = 1 and where
the difference td−dtends to infinity asd→ ∞.More precisely:
Theorem 13 ([3, Theorem 1]). Let d ∈ N with d ≥ 3 and let k be a non negative integer with 2k ≤d−3. There exists a sequence of d-generated finite soluble groups Jm with βd+k(Jm) ≥1−ϵm
for a certain sequence ϵm such thatϵm tends to0 as m tends to infinity.
Corollary 14. Letd∈Nwithd≥3and letF be the free prosoluble group of rankd.Thenβd+k(F) = 1
for every k such that2k≤d−3.
The groups described in [3] have a quite intricate structure and one would like to produce easier examples. By Proposition 7 these cannot be obtained just considering pronilpotent groups. However in [4] it is proved that if Gis the free prosupersoluble group of rankd≥2,thenPG(t)>0 if and only
ift≥2d+ 1 so one could expect to have βd+k(G) = 1 for ksignificantly larger than d.However this
is not what occurs. In fact we have:
Theorem 15([4, Theorem 2]). IfGis a noncylic finite supersoluble group andk≥3,thenβd(G)+k(G)≤
0.6
8. Positively unbiased groups
A profinite group G is called positively finitely generated (PFG) if PG(k) > 0 for some k ∈ N.
This concept actually first arose in the context of field arithmetic. Various theorems that are valid for “almost all” k-tuples in the absolute Galois group G(F) of a field F appear in [6]. Answering a question of Fried and Jarden [6], Kantor and Lubotzky [8] have shown that the free profinite group of rank d is not PFG if d ≥ 2. On the other hand, Mann [14] has proved that finitely generated prosoluble groups have this property. Denote bymn(G) the number of index nmaximal subgroups of
G. A groupG is said to have polynomial maximal subgroup growth (PMSG) ifmn(G)≤nc for all n
(for some constantc). A one-line argument shows that PMSG groups are positively finitely generated. By a very surprising result of Mann and Shalev [15] the converse also holds: a profinite group is PFG if and only if it has polynomial maximal subgroup growth.
We introduce a similar notion in relation with the study of βk(G).We say that a finitely generated
profinite group is positively unbiased generated (PUG) is βk(G)<1 for some k∈N. It follows from
Corollary 9 that ifPG(t)>0 thenβt(G)<1,so PFG groups are PUG. However, as we noticed at the
end of the Section 6, it can be that βt(G)< 1 even ifPG(t) = 0.On the other hand we don’t know
Question 2. Is it true that a profinite groups Gis PFG if and only if it is PUG?
9. Some Frobenius groups
In the remaining part of this survey, we compute βt(G) in other significant families of finite groups.
We start considering some Forbenius groups. Assume that p and q are prime numbers and leta be the multiplicative order of pmodule q.The affine group Aff(1, pa) contains a subgroupG∼=Cpa⋊Cq.
The socle N ofGis isomorphic toCpaand is the unique minimal normal series ofG. We can compute PG(g, t) applying formula (2.3) to the chief series 1< N < G.
(1) If g= 1,then, since N haspa complements in G, we have
PG(g, t) =PG(t) =
(
1− p
a
pat
) (
1− 1
qt
)
.
(2) If |g|=p, then there is no complement ofN inG containingg hence
PG(g, t) =
(
1− 1
qt
)
.
(3) If |g|=q, then ⟨g⟩N =G and there is no complement of G/N inG/N containing gN, while
⟨g⟩ is the unique complement of N inGcontainingg. Therefore
PG(g, t) =
(
1− 1
pat
)
.
It follows that for t≥2 we have
σG,t(g, t) =
(pat−p2a) (pat−pa)
(qt−q)
(qt−1) if|g|= 1,
σG,t(g, t) =
pat
(pat−pa)
(qt−q)
(qt−1) if|g|=p,
σG,t(g, t) =
qt
qt−1 if|g|=q.
Notice that pat(qt−q) ≤ (pat−pa)(qt−1) if and only (qt−1)/(q−1) ≤pa(t−1). Since q ≤ pa−1 we have (qt−1)/(q−1)≤(q+ 1)t−1 ≤pa(t−1).But thenσ
G,t(g, t) ≤1 if and only if g∈N and this
implies
βt(G) =βt(G/N) =βt(Cq) =
(q−1) (qt−1)q.
In the particular case when p= 3 andq = 2, we get
βt(Sym(3)) =βt(C2) = 1
10. G= Sym(4)
If G= Sym(4), we have
PG(t) =
(
1− 1
2t
) (
1− 3
3t
) (
1− 4
4t
)
.
We computeσG,t(g) =PG(g, t−1)/PG(t) wheregbelongs to a set of representatives for the conjugacy
classes of G:
σG,t(1) =
(
1− 2t1−1 ) (
1− 3t3−1 ) (
1− 4t4−1 )
PG(t)
= (2
t−2)(3t−9)(4t−16)
(2t−1)(3t−3)(4t−4)
σG,t((1,2)(3,4)) =
(
1− 2t1−1 ) (
1− 3t3−1 )
PG(t)
= (2
t−2)(3t−9)4t
(2t−1)(3t−3)(4t−4)
σG,t((1,2,3)) =
(
1− 2t1−1 ) (
1− 4t1−1 )
PG(t)
= (2
t−2)3t(4t−4)
(2t−1)(3t−3)(4t−4)
σG,t((1,2)) =
(
1− 1 3t−1
) (
1− 2 4t−1
)
PG(t)
= 2
t(3t−3)(4t−8)
(2t−1)(3t−3)(4t−4)
σG,t((1,2,3,4)) =
(
1− 3t1−1 )
PG(t)
= 2
t(3t−3)4t
(2t−1)(3t−3)(4t−4).
Notice that
• σG,t(1)<1 for everyt≥2;
• σG,t((1,2)(3,4))<1 for every t≥2;
• σG,t((1,2,3))<1 for every t≥2;
• σG,t((1,2))>1 for every t≥3 while σG,2((1,2,3))<1;
• σG,t((1,2,3,4))>1 for every t≥2.
It t≥3,thenσG,t(g)≤1 if and only if g∈Alt(n) and therefore
βt(Sym(4)) =βt(Sym(4)/Alt(4)) =βt(C2) = 1 (2t−1)2.
However if t= 2 thenσG,t(g)>1 only ifg is a 4-cycle and
β2(Sym(4)) = 6 24
(
22 22−1
42 42−4−1
)
= 7
36 > β2(C2) = 1 6.
11. G= Sym(n)
We have seen in the previous two examples that if G∈ {Sym(3),Sym(4)} andt≥3,then βt(G) =
βt(C2).The situation is similar for all the symmetric groups. By (2.2), for g∈G= Sym(n),we have PG(g, t) =
∑
n∈N
a(n, g)
nt with a(n, g) =
∑
g∈H,|G:H|=n
Since Alt(n) is the unique subgroup of Sym(n) of index 2 we have
PG(g, t) =
1− 1 2t +
∑
n≥3
a(n,g)
nt ifg∈Alt(n)
1 +∑
n≥3
a(n,g)
nt otherwise.
This implies that if tis large enough then
σG,t(g)∼
2t−2
2t−1 <1 ifg∈Alt(n)
2t
2t−1 >1 otherwise.
Therefore, by Proposition 3, if tis large enough, we have
βt(Sym(n)) =βt
(
Sym(n) Alt(n)
)
=βt(C2) =
1 (2t−1)2.
12. Asymptotically unbiased normal subgroups
We say that a normal subgroup N of a finite group G is asymptotically unbiased if there exists t∈Nsuch that βt(G) =βt(G/N) for allt≥t.
Let l(G) = minH<G|G : H| and for each g in G denote by ρ(G, g) the number of (maximal)
subgroups ofG of indexl(G) containingg. It follows from (2.2) that
σG,t(g) =
PG(g, t−1)
PG(g)
=
1−l(Gl)(ρG()G,gt ) +o (
1
l(G)t )
1−ρl((G,G)1)t +o (
1
l(G)t ) .
This implies
• ifρ(G,1)< l(G)ρ(G, g) theng∈∆−G(t) for tlarge enough;
• ifρ(G,1)> l(G)ρ(G, g) theng∈∆+G(t) for tlarge enough.
Corollary 16. If N is asymptotically unbiased, thenρ(G,1)≤l(G)ρ(G, n) for each n∈N.
Proof. Notice that σG,t(1) < 1 for each t ∈ N. Since 1 ∈ N, we deduce from Proposition 3 that if
βt(G) =βt(G/N) thenn /∈∆+G(t) for eachn∈N. □
Proposition 17. Assume that |G :N| = 2. Then N is asymptotically unbiased if and only if N is the unique subgroup of G of index 2.
Proof. Assume thatN is the unique subgroup ofGof index 2. We havel(G) = 2,ρ(G, g) = 1 ifg∈N, ρ(G, g) = 0 otherwise. This implies that if t is large enough, then g ∈ ∆−G(t) if g ∈ N, g ∈ ∆+G(t) if g /∈ N and the conclusion follows from Proposition 3. On the other hand, if N is not the unique subgroup of G of index 2, then there exists M ⊴G such that G/M ∼=C2×C2.By Proposition 6 we deduce
βt(G)≥βt(G/M) =βt(C2×C2) =
p2−1 (pt−1)p2 >
p−1
(pt−1)p =βt(C2) =βt(G/N)
13. G= Alt(5)
We compute PG(g, t) using the formula (2.2). The subgroupsH of GwithµG(H)̸= 0 are listed in
Table 2.
Table 2.
H Alt(5) Alt(4) D10 Sym(3) C3 C2 1 µG(H) 1 -1 -1 -1 2 4 -60
We have in particular
PG(1, t) = 1−
5 5t −
6 6t −
10 10t+
20 20t +
60 30t −
60 60t
PG((1,2,3,4,5), t) = 1−
1 6t
PG((1,2,3), t) = 1−
2 5t −
1 10t+
2 20t
PG((1,2)(3,4), t) = 1−
1 5t −
2 6t −
2 10t+
4 30t.
It can be easily checked that for t ≥2 we have that PG(g, t−1)≥ PG(t) if and only if |g|= 5 and
therefore by Proposition 3
βt(G) =
24 60
(
1− 1 6t−1
PG(t) −
1
)
= 24
(5
5t +1010t − 2020t −3060t +6060t )
60(
1−55t −66t −1010t +2020t +3060t −6060t ).
In particularβ2(Alt(5)) = 12/95.
14. Other simple groups
If S is a nonabelian finite simple group, thend(G) = 2 and results of Dixon [5], Kantor-Lubotzky [8] and Liebeck-Shalev [9] establish thatPS(2)→1 as |S| → ∞.In a recent paper [16] it was proved
that PS(2)≥53/90 for each nonabelian simple groupS.Since βS(2)≤1−PS(2),we deduce that
β2(S)≤47/90
and β2(S)→0 as|S| → ∞.In [16] it is also proved that the equalityPS(2) = 53/90 is satisfied if and
only if S = Alt(6).It is a natural question to ask which is the largest value for β2(S) and for which simple group this value is assumed. We have seen in the previous section thatβ2(Alt(5)) = 12/95.We check whether there exists a finite nonabelian simple groupT ̸= Alt(5) withβ2(T)≥12/95.We should have 12/95≤β2(T)≤1−PT(2) hencePT(2)≤83/95: by [16, Theorem 1.1],T is one of the following
groups: Alt(5),Alt(6),Alt(7),Alt(8),Alt(9),Alt(10), L2(7), L2(8), L2(11), L3(3), L3(4), M11, M12.The behavior of these groups is described in Table 3. From this table we conclude that does not exist such a nonabelian finite simple group being non-isomorphic to Alt(5). So we have proved:
This gives another evidence that the estimations for βt(G) deduced from the inequality βt(G) ≤
1−PG(t) are not sharp. Since PS(2) assumes its minimal value when S = Alt(6), one could expect
that the maximus value for βt(G) is assumed again when G = Alt(6). On the contrary, we get
β2(Alt(5))> β2(Alt(6)).
Table 3.
T β2(T) PT(2)
Alt(5) 12/95∼0.126 19/30∼0.633 Alt(6) 131/1060∼0.124 53/90∼0.588 Alt(7) 1469/19236∼0.076 229/315∼0.726 Alt(8) 4027/55860∼0.072 133/180∼0.738 Alt(9) 5314873/99811440∼0.053 15403/18144∼0.849 Alt(10) 6509536/138919725∼0.047 29401/33600∼0.875 L2(7) 181/1596∼0.113 19/28∼0.678 L2(11) 338/4191∼0.081 127/165∼0.769 L2(8) 40/639∼0.063 71/84∼0.845 L3(3) 243/5252∼0.047 101/117∼0.863 L3(4) 1021/25410∼0.040 121/140∼0.864 M11 24223/427548∼0.057 3239/3960∼0.818 M12 30851/708840∼0.044 179/220∼0.814
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Eleonora Crestani
Dipartimento di Matematica , Via Trieste 63 , 35121 Padova , Italy
crestani@math.unipd.it
Andrea Lucchini
Dipartimento di Matematica , Via Trieste 63 , 35121 Padova , Italy
