arXiv:1711.01881v3 [math.LO] 25 Feb 2018

FINITE RANK AND PSEUDOFINITE GROUPS
DANIEL PALACÍN

Abstract. It is proven that an inﬁnite ﬁnitely generated group cannot
be elementarily equivalent to an ultraproduct of ﬁnite groups of a given
Prüfer rank. Furthermore, it is shown that an inﬁnite ﬁnitely generated
group of ﬁnite Prüfer rank is not pseudoﬁnite.

1. Introduction
A pseudofinite group is an infinite model of the first-order theory of finite
groups, or equivalently, an infinite group is pseudofinite if it is elementary
equivalent to an ultraproduct of finite groups. In a similar way, one can
define pseudofinite fields, which were first considered by Ax [1]. These two
notions are closely related as exhibited by Wilson in [17], who showed that
a simple pseudofinite group is elementarily equivalent to a Chevalley group
over a pseudofinite field. Furthermore, it was later observed by Ryten [13]
that it is even isomorphic to such a group, see [3, Proposition 2.14(i)] for a
more detailed argument.
In a pseudofinite group any definable map is injective if and only if it is
surjective. Hence, it can be easily noticed, by considering the definable map
x 7→ n · x for a suitable positive integer n, that an infinite finitely generated
abelian group is not pseudofinite. Consequently, since centralizers of nonidentity elements in free groups are infinite cyclic groups, free groups, and
in particular finitely generated ones, are not pseudofinite. In fact, Sabbagh
raised the following question, which so far remains open:
Question. Are there infinite finitely generated groups which are pseudofinite?
One can of course reformulate this question and ask which families of finite
groups admit an ultraproduct elementarily equivalent to an infinite finitely
generated group. Using deep results from finite group theory, Ould Houcine
and Point [11] have made substantial progress in this direction and towards
the understanding of the structure of a possible example. They showed
Date: February 27, 2018.
2010 Mathematics Subject Classification. 20A05, 03C60.
Key words and phrases. Pseudoﬁnite group, Finite generation, Prüfer rank.
Research partially supported by the program MTM2014-59178-P and the European
Research Council grant 338821.
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DANIEL PALACÍN

that an infinite finitely generated pseudofinite group cannot be solvable-by(finite exponent), generalizing a previous result of Khélif, see [11, Proposition
3.14] for more details. In particular, an infinite finitely generated cannot be
elementarily equivalent to a family {Gi }i∈N of finite groups such that the
solvable radical R(Gi ) of Gi has a given derived length and that Gi /R(Gi )
has a fixed finite exponent.
The aim of this note is to obtain a similar result for another well-understood
family of finite groups, finite groups of bounded Prüfer rank. Recall that a
group is said to have finite Prüfer rank r if every finitely generated subgroup
can be generated by r elements, and r is the least such integer. We prove
the following:
Theorem 1. There is no infinite finitely generated group which is elementarily equivalent to an ultraproduct of finite groups of a given Prüfer rank.
This is deduced from a slightly stronger result where the assumption on
the Prüfer rank is relaxed, see Proposition 3.5. There, it is assumed that
only specific quotients have finite rank. As a consequence from this stronger
result, we immediately obtain in Corollary 3.6 that there is no infinite finitely
generated pseudofinite group of finite Prüfer rank.
2. Preliminaries on finite and pseudofinite groups
Let G be an arbitrary group. Given a subset X of G and a positive integer
k, we write
X ∗k = {x1 . . . xk : x1 , . . . , xk ∈ X}
and denote by hXi the subgroup of G generated by X, i.e.
[
(X ∪ X −1 )∗k .
hXi =
k∈N

Given a word w(x̄) in the variables x̄ = (x1 , . . . , xm ), let w(G) denote the
set of elements w(ḡ) of G for m-tuples ḡ from G. As w(G) is invariant under
Aut(G), the verbal subgroup hw(G)i is characteristic but not necessarily
definable in the pure language of groups. The width of the word w(x̄) in
G is the smallest integer k, if it exists, such that hw(G)i = w(G)∗k . In
particular, if w(x̄) has finite width in G, then the verbal group generated by
w(G) is clearly definable.
We focus our attention on two families of words: The family of words
wn (x) = xn of nth powers for each positive integer n and the family of simple
commutator words wc,n (x1 , . . . , x2n ) defined recursively on n as follows
−1
wc,1 (x1 , x2 ) = [x1 , x2 ] = x−1
1 x2 x1 x2

and


wc,n+1 (x1 , . . . , x2n+1 ) = wc,n (x1 , . . . , x2n ), wc,n (x2n +1 , . . . , x2n+1 ) .

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In a series of papers, Nikolov and Segal obtained deep results concerning
the finite width of certain families of words. Of special interest for our
purposes are [8, Theorem 1.7] (see also [10, Theorem 1.2]) and [9, Theorem
2], where they show the existence of positive integers δc (d) and δn (d) such
that the words wc,1 and wn have width δc (d) and δn (d) in any d-generator
finite group, respectively. In [11], Ould Houcine and Point noticed that
these results transfer to pseudofinite groups by showing that these words
have finite width in any group that is elementarily equivalent to a nonprincipal ultraproduct of d-generator finite groups, see Proposition 3.7 there.
As a consequence, the verbal subgroups generated by wc,1 (G) and wn (G) are
definable. Furthermore, they proved the following in [11, Proposition 3.12]:
Fact 2.1. Let G be a finitely generated pseudofinite group. Then the words
wc,n and wn have finite width in G. Moreover, the subgroups G(n) = hwc,n (G)i
and Gn = hwn (G)i are definable and have finite index.
It is worth noticing that Gn has finite index in G by Zelmanov’s positive
solution to the restricted Burnside problem. Next, using the fact above we
note the following:
Lemma 2.2. Let G be a finitely generated pseudofinite group and assume
that it contains an infinite definable definably simple non-abelian normal
subgroup N . Then, the group N is a finitely generated pseudofinite group.
Here, by a definably simple group we mean a group that has no proper nontrivial normal subgroups definable in the pure language of groups.
Proof. Let G be a finitely generated pseudofinite group G and assume that it
contains an infinite definably simple non-abelian normal subgroup N which
is definable by some formula φ(x, ā). By Frayne’s Theorem [2, Corollary
4.3.13], there exists some non-principal ultraproduct of an infinite family
{Gi }i∈I of finite groups in which G is elementarily contained. Let āi be a
representative of the equivalence class ā in Gi and let Ni be the set φ(Gi , āi ).
Since by Łos’s Theorem, almost every subset Ni is a normal subgroup of Gi ,
we may assume that every subset Ni is indeed a normal subgroup of Gi .
Furthermore, as N is elementary equivalent to an ultraproduct of the family
{Ni }i∈I , we may assume by [15, Corollary 2.10] that every subgroup Ni is a
finite simple group.
The action of Gi by conjugation on each Ni gives rise to an embedding from
Gi /CGi (Ni ) into the group of automorphisms Aut(Ni ) of Ni , yielding a subgroup containing the group of inner automorphisms Inn(Ni ) of Ni . It is clear
that Inn(Ni ) is isomorphic to Ni since the latter is centerless. Furthermore,
we known by [4, Chapter 4.B page 304] that the group Aut(Ni )/Inn(Ni ) is
solvable of derived length three, and then so is each Gi /(Ni CGi (Ni )).
Now, since the normal subgroup Ni CGi (Ni ) is definable over āi , it can be
expressed as a first-order property of the tuple āi that Gi modulo Ni CGi (Ni )

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DANIEL PALACÍN

is solvable of derived length three. Namely, it suffices to express that any
element of Gi of the form wc,3 (x1 , . . . , x8 ) belongs to Ni CGi (Ni ). Hence, we
obtain by Łos’s Theorem that the pseudofinite group G/(N CG (N )) is also
solvable of derived length three and so it is finite by Fact 2.1. Whence, the
group N CG (N ) is finitely generated and so is N CG (N )/CG (N ), which is
clearly isomorphic to N , yielding the result.

Another crucial point in our proofs, which also relies on the work of Nikolov
and Segal [8, 10], is a result due to Wilson [18].
Fact 2.3. There exists a formula φs (x) in the pure language of groups that
defines the solvable radical of any finite group.
It is clear that Wilson’s formula induces a characteristic subgroup in any
pseudofinite group. A priori this subgroup might not be solvable as witnessed by an ultraproduct of finite solvable groups without a common bound
on their derived lengths. Nevertheless, as a definable factor of a pseudofinite group is also pseudofinite, see [11, Lemma 2.16], the existence of such
formula allows us to split a pseudofinite group into a semisimple part and
a solvable-like one. Here, by a semisimple group we mean a group without non-trivial abelian normal subgroups. In contrast with being simple,
notice that semisimplicity is an elementary property. Namely, a group G
is semisimple if and only if it satisfies a first-order sentence asserting that
there is no non-trivial element which commutes with all its conjugates, since
for any element a the abelian normal subgroup Z(CG (aG )) of G is always
definable.
3. Proof of the Theorem
We first need a lemma on finite simple groups, which of course may be wellknown. Our proof is inspired by the one of [7, Lemma 4.2].
Lemma 3.1. If S is a finite simple non-abelian group of finite 2-rank r, then
either |S| is r-bounded or S is a finite simple group of Lie type of r-bounded
Lie rank over a finite field.
Here and throughout the paper, by r-bounded we mean that there is a bound
depending only on r.
Proof. We know by the Classification of Finite Simple Groups that a finite
simple non-abelian group is either an alternating group, a simple group of
Lie type or one of the finitely many sporadic groups.
It is clear that we can bound the size of all sporadic groups. Moreover, since
the alternating group Alt(n) contains an elementary abelian 2-subgroup of
rank 2 · ⌊n/4⌋, the size of an alternating group of 2-rank r is also r-bounded.
Suppose that S is a finite simple group of Lie type of Lie rank n over the
finite field Fq and assume further that it has 2-rank r. To bound the Lie rank

FINITE RANK AND PSEUDOFINITE GROUPS

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in terms of r, it suffices to assume that S is a classical simple group. In that
case, we have that S contains PSLm (Fq ), where m = max{2, ⌊ 21 (n − 1)⌋}.
Thus, since Alt(m) embeds of into PSLm (Fq ) via permutation matrices, the
paragraph above implies that m and hence n are r-bounded.

Lemma 3.2. A semisimple pseudofinite group of finite 2-rank has an infinite
definable non-abelian simple normal subgroup which is a linear group.
Proof. Assume that G is a semisimple pseudofinite group of 2-rank at most
r. Thus, it is elementarily equivalent to an ultraproduct of an infinite family
{Fi }i∈I of finite groups with respect to a non-principal ultrafilter U on the set
I. After shrinking I and U if necessary, we assume that all Fi are semisimple
and of 2-rank at most r. Notice that the latter property is clearly expressible
by a first-order sentence asserting the non existence of r +1 elements of order
2 which commute pairwise.
Now, set Soc(Fi ) to denote the socle of Fi , the subgroup of Fi generated by
all minimal normal subgroups. Thus
Soc(Fi ) = Si,1 × . . . × Si,ki ,
where each Si,j is a non-abelian simple normal subgroup of Fi . Note that for
every i we have ki ≤ r by the Feit-Thompson Theorem, since each Si,j contains an element of order 2 and any two Si,j and Si,l commute. Furthermore,
by Lemma 3.1 there is a positive integer δ(r), depending only on r, such that
either |Si,j | ≤ δ(r) or Si,j is a simple group of Lie type of Lie rank at most
δ(r). On the other hand, as we are assuming that G is infinite, the size of the
groups Fi is unbounded with respect to U . Hence, since CFi (Soc(Fi )) = 1,
the group Fi acts faithfully by conjugation on Soc(Fi ) and so it embeds into
the group of automorphisms of Soc(Fi ). Thus, the size of the socle Soc(Fi )
must be unbounded with respect to U as well. Consequently, as there is only
a finite number of possible Lie ranks, there is some τ ≤ δ(r) and a subset
I0 in U such that for each i ∈ I0 some Si,ji is a simple group of a fixed Lie
type of Lie rank τ and moreover, the size of all these Si,ji is unbounded with
respect to U .
Next, we see that every Si,ji with i ∈ I0 can be defined in a uniform way.
To do so, we use [6, Theorem 4.2], which asserts the existence of a positive
integer m = m(τ ) such that any finite simple group of a fixed Lie type of
Lie rank τ is equal to the set (C ∪ C −1 )∗m for any non-trivial conjugacy
class C with |C| > m. Hence, using this result, we find a formula ψ(x, y) in
the language of groups such that for some non-trivial element ai in Si,ji we
have that ψ(Fi , ai ) = Si,ji and such that the Si,ji -conjugacy class Ci of any
non-trivial element of Fi satisfying ψ(x, ai ) generates Si,ji in m many steps,
i.e. Si,ji = (Ci ∪ Ci−1 )∗m , provided that |Ci | > m.
By Łos’s Theorem, we can find an element a in G such that the subset ψ(G, a)
is an infinite definable normal subgroup of G and that ψ(G, a) = (C ∪C −1 )∗m
for any ψ(G, a)-conjugacy class C of every non-trivial element of ψ(G, a) with

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DANIEL PALACÍN

|C| > m. In fact, since ψ(G, a) is a definably simple group, any ψ(G, a)conjugacy class C of a non-trivial element of ψ(G, a) must be infinite, as
otherwise the centralizer of C in ψ(G, a) would be a definable (finite index)
normal subgroup, yielding a contradiction. As a consequence, it then follows
that the group ψ(G, a) is simple.
Finally, for the second part of the
Qstatement, note that ψ(G, a) is elementarily
equivalent to the ultraproduct
U Si,ji and so it is elementarily embeddable
Q
into an ultrapower of U Si,ji by Frayne’s Theorem [2, Corollary 4.3.13]. By
a result of Point [12], such an ultrapower is isomorphic to a simple group
of Lie type over a pseudofinite field. Hence, the simple group ψ(G, a) is a
linear subgroup.

Remark 3.3. In fact, the proof of the lemma yields the existence of a
simple group of Lie type by Ryten’s results. Namely, we have shown that
Q the
simple group ψ(G, a) is elementarily equivalent to the ultraproduct U Si,ji
of simple groups Si,ji of a fixed Lie type. Therefore, applying [3, Proposition
2.14(i)] we get that ψ(G, a) is a simple group of Lie type.
Before proving the main theorem, we deduce a weak version of Platonov’s
Theorem for pseudofinite groups.
Corollary 3.4. Any pseudofinite group of finite Prüfer rank is virtually
elementarily equivalent to an ultraproduct of finite solvable groups.
Proof. Let G be a pseudofinite group of finite Prüfer rank, and consider the
definable characteristic subgroup φs (G) of G. Set Ḡ to denote G/φs (G), a
semisimple group of finite Prüfer rank, and suppose that it is infinite. By
Lemma 3.2, we know that Ḡ contains an infinite simple normal subgroup
N which is a linear group. Hence, it is solvable-by-finite by Platonov’s
Theorem, see for instance [16, Theorem 10.9], and consequently it is finite, a
contradiction. Therefore, we obtain that Ḡ must be finite, which yields the
result by Fact 2.3.

In order to show Theorem 1, we shall state and prove the following slightly
stronger result. Before that, recall that the upper rank of a group is defined
as the maximum, if it exists, of the Prüfer rank of all its finite quotients.
Proposition 3.5. There is no finitely generated pseudofinite group G such
that G/φs (G) has finite 2-rank and that φs (G) has finite upper rank.
Proof. Suppose, towards a contradiction, that G is a finitely generated pseudofinite group witnessing a contradiction, and consider the definable characteristic subgroup φs (G) of G. Set Ḡ to denote G/φs (G), a finitely generated
semisimple group of 2-rank r.
The proof is divided in two parts:

FINITE RANK AND PSEUDOFINITE GROUPS

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First part (The semisimple case). We aim to prove that Ḡ is finite. Assume
otherwise that Ḡ is infinite. Thus, it contains an infinite definable simple
normal subgroup N which is as well a linear group by Lemma 3.2. Additionally, it is finitely generated by Lemma 2.2. Hence, by Malcev’s Theorem
it is residually finite, see for instance [16, Corollary 4.4], and consequently
it is finite, a contradiction. Therefore, we obtain that Ḡ must be finite, as
desired.
Second part (The solvable-like case). After the first part, the subgroup φs (G)
has finite index and so it is a finitely generated pseudofinite group of finite
upper rank. Hence, we may assume that G and φs (G) coincide, i.e. the
group G is elementarily equivalent to a non-principal ultraproduct of an
infinite family {Hi }i∈I of finite solvable groups.
The subgroups G(n) forming the derived series of G, as well as the subgroups
Gn generated by the nth powers, are all definable and have finite index in
G, by Fact 2.1. As each Hi is solvable and the words wc,1 have finite width
in G(n) also by Fact 2.1, an easy application of Łos’s Theorem yields that
G(n+1) is properly contained in G(n) . In particular, the group G has arbitrary
large finite quotients.
We claim that each finite quotient of G is solvable. To do so, consider a
finite index normal subgroup N of G and notice that it is definable, since
it contains the finite index subgroup Gk , where k = [G : N ], which is also
definable. Let χ(x, ȳ) be a formula in the pure language of groups such that
χ(G, c̄) = N for some finite tuple c̄ from G. By Łos’s Theorem, for almost
all indices i ∈ I (with respect to U ), we can find some tuple c̄i in Hi such
that χ(Hi , c̄i ) is a normal subgroup of Hi of index k. Hence, since each Hi
(k)
is solvable, the kth term Hi of the derived series of Hi must be contained
in χ(Hi , c̄i ) and so G(k) is contained in N by Łos’s Theorem.
To conclude, consider the characteristic subgroup G0 of G defined as the
intersection of all finite index normal subgroups of G. Note that each finite
quotient of G/G0 is solvable and has Prüfer rank at most r by assumption.
Thus, using the main result from [14] we obtain that G/G0 is nilpotent-byabelian-by-finite. Therefore, the quotient G/G0 is solvable and hence it must
be finite, a contradiction. This finishes the proof.

As an immediate consequence we deduce the following:
Corollary 3.6. There is no finitely generated pseudofinite group of finite
Prüfer rank.
We finally give the proof of the main theorem.
Proof of Theorem 1. Let G be an infinite finitely generated group and suppose towards a contradiction that there is an infinite family {Gi }i∈I ofQfinite
groups of Prüfer rank r such that G is elementarily equivalent to U Gi

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DANIEL PALACÍN

for some non-principal ultrafilter U on I. Consider Wilson’s formula φs (x).
Note that Gi /φs (Gi ) has 2-rank at most r and then, so does the pseudofinite
group G/φs (G) by Łos’s Theorem. Thus, the previous proposition yields
that φs (G) has finite index in G. Hence, after replacing G by φs (G) if necessary, we may assume that each Gi is a solvable group of Prüfer rank at
most r. Moreover, as remarked in the solvable-like case of the proof of the
previous proposition, we have that every finite index subgroup N of G is
definable, and so G has upper rank at most r. Therefore, applying once
more Proposition 3.5 we obtain the desired contradiction.


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Einstein Institute of Mathematics, The Hebrew University of Jerusalem,
Givat Ram 9190401, Jerusalem, Israel
E-mail address: daniel.palacin@mail.huji.ac.il

