Finite axiomatizability of the rank and the dimension of a pro-$\pi$ group

The Pr\"ufer rank $\mathrm{rk}(G)$ of a profinite group $G$ is the supremum, across all open subgroups $H$ of $G$, of the minimal number of generators $\mathrm{d}(H)$. It is known that, for any given prime $p$, a profinite group $G$ admits the structure of a $p$-adic analytic group if and only if $G$ is virtually a pro-$p$ group of finite rank. The dimension $\dim G$ of a $p$-adic analytic profinite group $G$ is the analytic dimension of $G$ as a $p$-adic manifold; it is known that $\dim G$ coincides with the rank $\mathrm{rk}(U)$ of any uniformly powerful open pro-$p$ subgroup $U$ of $G$. Let $\pi$ be a finite set of primes, let $r \in \mathbb{N}$ and let $\mathbf{r} = (r_p)_{p \in \pi}, \mathbf{d} = (d_p)_{p \in \pi}$ be tuples in $\{0, 1, \ldots,r\}$. We show that there is a single sentence $\sigma_{\pi,r,\mathbf{r},\mathbf{d}}$ in the first-order language of groups such that for every pro-$\pi$ group $G$ the following are equivalent: (i) $\sigma_{\pi,r,\mathbf{r},\mathbf{d}}$ holds true in the group $G$, that is, $G \models \sigma_{\pi,r,\mathbf{r},\mathbf{d}}$; (ii) $G$ has rank $r$ and, for each $p \in \pi$, the Sylow pro-$p$ subgroups of $G$ have rank $r_p$ and dimension $d_p$. Loosely speaking, this shows that, for a pro-$\pi$ group $G$ of bounded rank, the precise rank of $G$ as well as the ranks and dimensions of the Sylow subgroups of $G$ can be recognized by a single sentence in the first-order language of groups.


Introduction
Nies, Segal and Tent [Nies et al. 2021] carried out an investigation of the modeltheoretic concept of finite axiomatizability in the context of profinite groups.For instance, a profinite group G is finitely axiomatizable within a class C of profinite groups, with respect to the first-order language L gp of groups, if there is a sentence ψ G,C in L gp such that the following holds: a profinite group H in C is isomorphic to G if and only if ψ G,C holds true in H , in symbols H | ψ G,C .More generally, one takes interest in whether specific properties or invariants of profinite groups, again within a given class C, can be detected uniformly by a single sentence in L gp .
Our main interest is in finitely generated profinite groups.Nikolov and Segal [2007] established that such groups are strongly complete; loosely speaking, this means that the topology of a finitely generated profinite group is already predetermined by the abstract group structure.Jarden and Lubotzky [2008] used Nikolov and Segal's finite width results for certain words to prove that every finitely generated profinite group is "first-order rigid", i.e., determined up to isomorphism by its first-order theory, within the class of profinite groups.By restricting to finite axiomatizability, we probe for more delicate first-order properties within suitable classes of finitely generated profinite groups.
In this paper we focus on the class of profinite groups of finite Prüfer rank, from now on "rank" for short.This invariant is connected to, but not to be confused with, the minimal number of generators: the rank of a profinite group G is defined as where d(H ) denotes the minimal number of generators of a topological group H and, as indicated, H runs over all open or all closed subgroups of G.It is not difficult to see that the rank of G is the supremum of the ranks of its finite continuous quotients, i.e., rk(G) = sup{rk(G/N ) | N ⊴ o G}.The rank plays a central role in the structure theory of p-adic Lie groups.It is known that, for any given prime p, a profinite group G admits the structure of a p-adic analytic group if and only if G is virtually a pro-p group of finite rank.The dimension dim G of a p-adic analytic profinite group G is the analytic dimension of G as a p-adic manifold; in fact, dim G ≤ rk(G) and dim G coincides with the rank rk(U ) of any uniformly powerful open pro-p subgroup U of G. Further details and related results about p-adic analytic pro-p groups can be found in [Dixon et al. 1999]; the concise introduction [Klopsch 2011] summarizes key aspects of the theory.
Loosely speaking, our aim is to show that, for every finite set of primes π, the precise rank r as well as the ranks r = (r p ) p∈π and dimensions d = (d p ) p∈π of the Sylow pro-p subgroups of any pro-π group G of finite rank can be recognized by a single sentence σ π,r,r,d in the first-order language of groups L gp .The starting point for our investigation is Proposition 5.1 in [Nies et al. 2021] which states: Given r ∈ ‫,ގ‬ there is an L gp -sentence ρ p,r such that for every pro-p group G, the following implications hold: Our first theorem both strengthens and generalizes this result.The p-rank rk p (G) of a profinite group G is the common rank of all Sylow pro-p subgroups of G.A sentence φ in L gp is called an ∃∀∃-sentence if it results from a quantifier-free formula φ 0 by means of a sequence of existential, universal and existential quantifications (in this order), rendering the free variables of φ 0 to be bound in φ; compare with Example 3.1.
Theorem 1.1.Let π be a finite set of primes.Let r ∈ ‫ގ‬ and let r = (r p ) p∈π be a tuple in {0, 1, . . ., r }.Then there exists an ∃∀∃-sentence ϱ π,r,r in L gp such that, for every pro-π group G, the following are equivalent: (i) rk(G) = r , and rk p (G) = r p for every p ∈ π .
It is no coincidence that the sentences ϱ π,r,r which we manufacture to prove the theorem depend on the given set of primes π .A standard ultraproduct construction reveals that, for every infinite set of primes π and r ∈ ‫,ގ‬ there is no L gp -sentence ϑ π ,r which could identify, uniformly across p ∈ π, among pro-p groups G those with rank rk(G) = r ; see Proposition 3.3.
In addition to Theorem 1.1 we establish a corresponding theorem which concerns the dimensions of the Sylow subgroups of a profinite group of finite rank.
Theorem 1.2.Let π be a finite set of primes.Let r ∈ ‫ގ‬ and let d = (d p ) p∈π be a tuple in {0, 1, . . ., r }.Then there exists an ∃∀∃-sentence τ π,r,d in L gp such that, for every pro-π group G with rk(G) = r , the following are equivalent: (i) For every p ∈ π, the Sylow pro-p subgroups of G have dimension d p .
In combination, the two theorems provide the first-order sentences σ π,r,r,d with the properties promised above.It is remarkable that such sentences exist in the basic language L gp of groups.In connection with p-adic analytic profinite groups, it is often necessary to employ suitably expanded languages in order to capture part of the topological or analytic structure; compare with [Macpherson and Tent 2016].We do not need to enlarge the language at all.Moreover, the complexity of σ π,r,r,d remains within three alternations of ∃-and ∀-quantifiers, even though the sentences that we manufacture depend strongly on the given set of primes π .
As we will show, the proofs of Theorems 1.1 and 1.2 reduce, in a certain sense, to the simpler setting of pronilpotent pro-π groups, termed C π -groups by [Nies et al. 2021, Section 5].We recall that, even in the pronilpotent case, Sylow subgroups are not in general definable and there is no standard reduction to pro-p groups; this can be seen from relative quantifier elimination results (down to positive primitive formulas) for modules over rings; see [Prest 1988, Sections 2.4 and ‫.]ޚ.2‬Part of our task is to develop appropriate tools to by-pass this obstacle.
Key to our approach for proving Theorems 1.1 and 1.2 are purely group-theoretic considerations leading to Theorem 2.1 and its corollary, about profinite groups which are virtually pronilpotent and of finite rank.Specialising to the setting of finite nilpotent groups to ease the exposition at this point, we can formulate the central insight as follows.
It is an open problem to identify, if at all possible, even smaller canonical quotients which witness the full rank of a finite nilpotent group.
Following a suggestion of González-Sánchez, we derive from a result of Héthelyi and Lévai [2003] a new description of the dimension of a finitely generated powerful pro-p group; this is useful for establishing Theorem 1.2, but also of independent interest.
Theorem 1.4.Let G be a finitely generated powerful pro-p group with torsion subgroup T , and let {1} (G) = {g ∈ G | g p = 1} denote the set of all elements of order 1 or p in G. Then With a view toward possible future investigations, we add a final comment and a question.Naturally one wonders whether "being of finite rank" per se can be captured by a suitable first-order sentence.Results of Feferman and Vaught [1959] imply that, even for a fixed prime p, there is no set T p of L gp -sentences (and in particular no single sentence) which identifies among the collection of all pro-p groups those that possess finite rank.Indeed, the class of pro-p groups of finite rank is closed under taking finite cartesian products, but an infinite cartesian product of nontrivial pro-p groups of finite rank is not even finitely generated.Therefore [Feferman and Vaught 1959, Corollary 6.7] shows that no T p with the desired property exists.However, a modified question suggests itself.Given d ≥ 2, is there a set T p,d of L gp -sentences (possibly a single sentence) such that the following holds for pro-p groups G with d(G) ≤ d: the group G has finite rank if and only if G satisfies T p,d ?Remark.Our proofs for Theorems 1.1 and 1.2 involve results of Lucchini [1997] and an observation of Mazurov [1994] which currently rely on the classification of finite simple groups.However, in suitable circumstances, e.g., if we restrict attention to prosoluble groups, the required ingredients are known to hold without use of the classification; compare with [Lucchini 1989, Section 5].If 2 ̸ ∈ π , the Odd Order Theorem guarantees that all pro-π groups are prosoluble.
Organization and Notation.In Section 2 we prove Theorem 2.1 and its corollary, which specialize to Theorem 1.3.In Example 2.3 we discuss limitations of our strategy; Proposition 3.3 shows that Theorem 2.1 does not generalize to groups involving infinitely many primes.In Section 3 we establish Theorem 1.1.In Section 4 we prove Theorem 1.4 and deduce Theorem 1.2.
Our notation is mostly standard and in line with current practice.For instance, Z(G) denotes the centre of a group G, and C n denotes a cyclic group of order n.The meaning of possibly less familiar terms, such as (G) for the Frattini subgroup and p (G) for the p-Frattini subgroup of a group G, are explained at their first occurrence.We deal exclusively with profinite groups.Accordingly, notions such as the Frattini subgroup, the commutator subgroup or the subgroup generated by a given set are tacitly understood in the topological sense: in each case we mean the topological closure of the corresponding abstract subgroup.Basic model-theoretic concepts which are employed without further reference are covered by standard texts such as [Hodges 1993].

Detecting the rank in bounded quotients
Every compact p-adic analytic group G has finite rank and contains an open normal powerful pro-p subgroup F. Since F is a pro-p group, its Frattini subgroup (F) coincides with [F, F]F p and F/ (F) is elementary abelian.Since F is powerful, we know that rk(F) = d(F) = rk(F/ (F)); see [Dixon et al. 1999, Theorem 3.8].Furthermore, the iterated Frattini series j (F), j ∈ ‫,ގ‬ of F coincides with both the lower p-series and the iterated p-power series of F. It provides a base of neighbourhoods for 1 in G consisting of open normal subgroups.Consequently, the rank of G is given by rk It is natural to look for an upper bound for the smallest j ∈ ‫ގ‬ such that rk(G) = rk(G/ j (F)), a bound that is, as far as possible, independent of p and any special features of the pair F ≤ G. Based on our current knowledge, the strongest possible outcome could be that rk(G) = rk(G/ (F)) holds without any exceptions.More modestly, one can ask for weaker bounds, possibly contingent on additional information regarding rk(G).
We establish a result of the latter kind, which applies more generally to profinite groups G of finite rank that admit a pronilpotent open normal subgroup F. We recall that the p-rank rk p (G) of a profinite group G is simply the rank rk(P) of a Sylow pro-p subgroup P of G. Furthermore, we write p (G) = [G, G]G p for the p-Frattini subgroup of G; the p-Frattini quotient G/ p (G) is the largest elementary abelian pro-p quotient of the profinite group G.
Theorem 2.1.Let R ∈ ‫.ގ‬ Suppose that the profinite group G has an open normal subgroup F ⊴ o G which is pronilpotent and such that each Sylow subgroup of F is powerful.
(i) For every prime p such that rk p (G) ≤ R, the p-rank satisfies Since F is pronilpotent, its Hall pro-p ′ subgroup P ′ is normal in G; compare with [Ribes and Zalesskii 2010, Section 2.3].Working modulo P ′ , we may assume without loss of generality that F is a powerful pro-p group.In this situation G is virtually a pro-p group.Clearly, we have r p ≥ rk p (G/F 2R+1 ).For a contradiction, we assume that The sequence d(HF j /F j ), j ∈ ‫,ގ‬ is nondecreasing and eventually constant, with final constant value d(H ).Since d(H ) = r p < 2R + 1, we conclude that d(HF j /F j ), j ∈ ‫,ގ‬ cannot be strictly increasing until it becomes constant.Hence there exists j = j (H ) ∈ ‫ގ‬ such that and there is no harm in assuming that Choose a 1 , . . ., a m ∈ A with b i = a p i for 1 ≤ i ≤ m and set H = ⟨y 1 , . . ., y l , a 1 , . . ., a m ⟩ ≤ G.
We claim that H is a p-subgroup of G such that Clearly, H ≤ HA is a p-group and H ⊆ H .Moreover, we see that HA = HA = L A. We may assume without loss of generality that G and there is no harm in assuming L ∩ A = 1.This gives We supplement y 1 , . . ., y l to a minimal generating set y 1 , . . ., y l , ã1 , . . ., ãn for the p-group H , for suitable n ∈ {0, 1, . . ., m} and ã1 , . where p is elementary abelian of rank mlr(G), and H/N acts via conjugation faithfully on N / p (N ) by power automorphisms (i.e., by nonzero homotheties if we regard N / p (N ) as an ‫ކ‬ p -vector space).
For short let us refer, somewhat effusively, within this proof to such a pair (H, N ) as a "runaway couple" for G with respect to p.
By (i), we have mlr(G) = mlr(G/F 2R+1 ), and hence it suffices to show: if G admits a runaway couple, then so does G/F 2R+1 , in fact, with respect to the same prime.Suppose that (H, N ) is a runaway couple for G with respect to an odd prime p so that H/ p (N ) ∼ = C q ⋉C r p p as detailed above, with the additional property that |G : H | is as small as possible.Assume for a contradiction that G/F 2R+1 does not admit a runaway couple.
As in the proof of (i) there is no harm in factoring out the Hall pro-p ′ subgroup P ′ of F, because H ∩ F ⊆ N and H ∩ P ′ ⊆ p (N ).Consequently we may as well assume that F ⊴ o G is a powerful pro-p group, which makes G virtually a pro-p group.
As in the proof of (i), the sequence is nondecreasing and eventually constant, with final constant value We use the same arguments as before to conclude that there exists j = j (H ) such that the analogue of (2-1) for H/ p (N ) holds and we reduce to the situation where [F j , F j ] = F 2 j = 1.This reduction renders G finite, with abelian normal p-subgroups A = F j and B = F j+1 = (F j ) = A p ; furthermore, we have It The normal subgroup M p (N ) ⊴ H decomposes as a direct product M × p (N ).
Recall that H/ p (N ) ∼ = C q ⋉ C r p p , with the action given by power automorphisms.

FINITE AXIOMATIZABILITY OF THE RANK & THE DIMENSION OF A PRO-π GROUP 263
We build a minimal generating set x, y 1 , . . ., y l , b 1 , . . ., b m for H modulo p (N ) by choosing x ∈ H ∖ N and y 1 , . . ., y l ∈ N which supplement b 1 , . . ., b m suitably.We set In this situation H = L M and we claim that L ∩ M = 1 so that Indeed, our construction yields that the intersection in These considerations show that the group H = L M maps onto and hence onto C q ⋉ C r p p .Thus H gives rise to a runaway couple for G, with respect to the prime p, just as H does.To conclude the proof we observe that The following corollary yields in particular Theorem 1.3 about finite nilpotent groups, which was showcased in the introduction for its succinctness.
Corollary 2.2.Let R ∈ ‫.ގ‬ Suppose that the profinite group G has an open normal subgroup F ⊴ o G which is pronilpotent.
Proof.As in the proof of Theorem 2.1, one reduces to the case in which F is a pro-p group for a single prime p. From rk(F) ≤ R it follows that ⌈log 2 (R)⌉+1 (F) ⊴ o G is powerful; compare with [Dixon et al. 1999, Chapter 2, Exercise 6].Thus we can apply Theorem 2.1 to ⌈log 2 (R)⌉+1 (F) in place of F. □ The following example puts the basic idea behind the proof of Theorem 2.1 into perspective.It indicates that one would need to take a different approach or at least make more careful choices in order to eliminate the dependency on the parameter R. Indeed, the example yields, for p i = a i a i+1 for 1 ≤ i < n, and a c n = a n .Here ‫ޚ‬ p denotes the additive group of the p-adic integers, viz. the infinite procyclic pro-p group.Then G = ⟨c, a 1 ⟩ is 2-generated, nilpotent of class n and has rank rk(G) = n + 1.For instance, which is unrelated to H , requires n + 1 generators, even modulo (F).

Finite axiomatizability of the rank
In this section we establish Theorem 1.1.We begin with a basic example which illustrates the concept of an ∃∀∃-sentence in L gp and related constructions which we use frequently; compare with [Nies et al. 2021, Sections 2 and 5].Despite its simplicity, the example is a key building block in later proofs, where we need to control the quantifier complexity of more involved first-order formulae.
Example 3.1.Let G be a profinite group and let N ⊆ G. Suppose that N is definable in G; this means that there is an L gp -formula ϕ(x), with a single free variable x, such that N = {g ∈ G | ϕ(g)}.

FINITE AXIOMATIZABILITY OF THE RANK & THE DIMENSION OF A PRO-π GROUP 265
Then the sentence ∃a 1 , . . .a n ∀x, y, z : can be used to express that N ⊴ G and G/N ∼ = B.The quantifier complexity of this sentence is the same as the quantifier complexity of ϕ increased by ∃∀.In particular, if N ⊆ c G is ∃-definable as a closed set, i.e., definable by means of an ∃-formula which implicitly ensures that N is topologically closed, we obtain an ∃∀∃-sentence to express that N ⊴ c G and G/N ∼ = B.
For instance, if we know or suspect that the commutator word has a certain finite width in G, we may consider the ∃-definable set for a given parameter r ∈ ‫,ގ‬ and formulate an ∃∀∃-sentence in L gp which expresses that, indeed, N is equal to the entire commutator subgroup [G, G] and that the abelianization G/[G, G] is isomorphic to a given finite group.
Sometimes we want to express, by means of an L gp -sentence, extra features of a definable subgroup H ≤ c G. This process typically involves quantification over elements of H rather than G which, in general, may increase the quantifier complexity of the resulting sentences.However, if H = {g ∈ G | ϕ(g)} is ∃-definable, where ϕ(x) takes the form ∃z : ϕ 0 (x, z) with ϕ 0 quantifier-free in free variables x and z 1 , . . ., z m , say, then H is "quantifier-neutral" in the following sense.Firstorder assertions about H can be translated into assertions of the same quantifier complexity about G, simply by expressing universal quantification over elements of H as ∀x, z : (ϕ 0 (x, z) → • • • ) and existential quantification over elements of H as ∃x, z : It is convenient to establish the assertions of Theorem 1.1 first for pronilpotent groups before considering the general situation.
Proof.We set k = |π|, write π = { p 1 , . . ., p k } and put q = q(π Similar to Example 3.1, there is an ∃∀∃-sentence β 1 in L gp to express that there are elements a 1 , . . ., a r in H such that every element h ∈ H can be written as h = r j=1 a e j j b, for suitable choices for e j ∈ {0, 1, . . ., q − 1} and We recall that d(H ) = d(H/ (H )) and that d(H ) ≤ r implies B(H ) = (H ); see [Dixon et al. 1999 et al. 1999, Proposition 2.6], it suffices to express that every commutator [x, y] of elements x, y ∈ F is a (2q)-th power z 2q of a suitable z ∈ F.
Once F is r -generated and semipowerful, we know that rk(F) ≤ r .If, in addition, the rank bounds specified in (3-1) hold, we deduce that rk(H/F) ≤ mr and hence rk(H ) ≤ R for R = (m + 1)r .Furthermore, the group is ∃-definable in H and hence quantifier-neutral; in particular, H/ 2R+1 (F) is interpretable in H . Finally, |H/ 2R+1 (F)| is bounded by q (2R+m+1)r and there is an ∃∀∃-sentence θ which expresses that H/ 2R+1 (F) is one of the finitely many finite π -groups of suitable order which has rank r and whose p-ranks are in agreement with the prescribed r; compare with Example 3.1.With the backing of Theorem 2.1, we form the conjunction of the sentences β m+1 , γ , θ to arrive at an ∃∀∃-sentence ω π,r,r with the desired property.□ Proof of Theorem 1.1.We analyse the structure of a pro-π group G of rank rk(G) = r to build step-by-step a first-order sentence η π,r that is satisfied by any such group G.
Following that we check that, conversely, every pro-π group satisfying η π,r has rank at most 2r .Applying Theorem 2.1, we extend η π,r to a sentence ϱ π,r,r which pins down precisely the rank as being r and the ranks of the Sylow subgroups as being given by r.
Our discussion involves upper bounds for certain integer parameters that depend on π and r , but not on the specific group G used in our discussion; for short, we say that such parameters are (π, r )-bounded.The proof proceeds in four steps along the following plan of action.In Step 1 we produce a pronilpotent open normal subgroup K ⊴ o G of (π, r )-bounded index.This is used in Step 2 to describe an ∃-definable pronilpotent open normal subgroup H ⊴ o G of (π, r )-bounded index.In Step 3 we show that the fact that H is pronilpotent can be expressed by an ∃∀∃-sentence.This uses a simple but effective trick: we would like to express that H is a direct product of its Sylow subgroups, but in general the latter fail to be definable; to overcome this problem we work modulo the centre Z(H ) which is sufficient for our purposes.In Step 4 we use the tools that we already prepared in Example 3.1 and in Proposition 3.2 to conclude the argument.
Step 1.The classification of finite simple groups implies that, up to isomorphism, there are only finitely many finite simple π-groups; see [Mazurov 1994, Remark following Lemma 2].A fortiori there is a finite set S = S π,r of representatives for the isomorphism classes of finite simple π-groups S such that rk(S) ≤ r .Consequently, the cardinality of the set = G,π,r = ψ | ψ : G → Aut(S l ) a homomorphism for S ∈ S and 0 ≤ l ≤ r is (π, r )-bounded, because G can be generated by at most r elements and any homomorphism between groups is determined by its effect on a chosen set of generators.From this we observe that the index of in G is (π, r )-bounded.Thus there exists f (π, r ) ∈ ‫,ގ‬ depending on π and r , but not on the specific group G, such that |G : K | divides f (π, r ).
We claim that K is pronilpotent.For this it suffices to show that K /(K ∩ L) is nilpotent for each L ⊴ o G. Let L ⊴ o G.By pulling back a chief series for the finite group G/L to G, we obtain a normal series of finite length n such that, for each i ∈ {1, . . ., n}, the group G i /G i+1 is a minimal normal subgroup of G/G i+1 and thus isomorphic to S m(i) i for suitable choices of S i ∈ S and m(i) ∈ ‫.ގ‬ Since each of the groups S m(i) i contains an elementary abelian p-subgroup of rank m(i), for primes p dividing |S i |, we obtain m(i) ≤ rk(S m(i) i ) ≤ rk(G) = r for all i ∈ {1, . . ., n}.Intersecting with K , we obtain a series satisfying 0 ≤ l(i) ≤ m(i) ≤ r , for i ∈ {1, . . ., n}.By construction, K acts trivially on each of these factors so that [K ∩G i , K ] ⊆ K ∩G i+1 for i ∈ {1, . . ., n}.Thus (3-2) constitutes a central series for K /(K ∩ L), and K /(K ∩ L) is nilpotent (of class at most n).
Step 2. Next we consider the group the index |G : H | is (π, r )-bounded, by the positive solution to the restricted Burnside problem.In fact, we do not require the general result, but a rather special case, which is easy to establish.Indeed, assume for the moment that the pro-π group G of rank r is finite of exponent f (π, r ).We need to show that |G| is (π, r )-bounded.In Step 1 we established that G has a nilpotent normal subgroup K of (π, r )-bounded index.Thus there is no harm in assuming that G = K .Furthermore, K is a direct product of its Sylow p-subgroups, where p ranges over the finite set π. Hence we may even assume that G is a p-group of rank at most r , for some p ∈ π , and that f (π, r ) is a p-power, p e say.In this situation, G contains a powerful normal subgroup of ( p, r )-bounded index (see [Dixon et al. 1999, Theorem 2.13]), and we may assume that G itself is powerful.The p-power series of a powerful p-group coincides with its lower p-series, and we obtain the bound |G| ≤ p r e .Next we observe that the verbal subgroup H is an ∃-definable subgroup of G and hence quantifier-neutral, in the sense discussed in Example 3.1.Indeed, by [Nikolov and Segal 2011, Theorem 1], every element of H can be written as a product of a (π, r )-bounded number of f (π, r )-th powers.But again we only require the bound in a rather special case which is much easier to handle.Indeed, descending without loss of generality to a subgroup of (π, r )-bounded index, as above, it suffices to recall that in a powerful pro-p group every product of p e -th powers is itself a p e -th power; see [Dixon et al. 1999, Corollary 3.5].
Step 3. Since K is pronilpotent, so is H .In the situation at hand, this fact can be expressed by an ∃∀∃-sentence.Indeed, H is pronilpotent if and only if H/ Z(H ) is pronilpotent.Hence it suffices to express the assertion that H we deduce that and thus As rk(G) ≤ r , there exist, for each i ∈ {1, . . ., k}, elements x i,1 , . . ., x i,r ∈ H i such that H i = ⟨x i,1 , . . ., x i,r ⟩ and thus Subject to the kr parameters x 1,1 , . . ., x k,r , this makes Z(H ) = k i=1 C i and each of the groups D i quantifier-free definable, by suitable centralizer conditions; moreover We conclude that it suffices to express in an ∀∃-sentence, subject to the (π, r )bounded number of parameters x s,t , that for this implies that H/ Z(H ) = k i=1 D i / Z(H ) is the direct product of its Sylow subgroups and thus pronilpotent.Turning the parameters x s,t into variables bound by an extra existential quantifier at the front, we arrive at an ∃∀∃-sentence without parameters which verifies that H is pronilpotent.
Subject to the parameters x s,t , the assertions in (a), (c) can be expressed by an ∀-sentence, and (d) can be achieved by means of an ∀∃-sentence.The only tricky part occurs in (b) where we need to express that the group Q i = D i / Z(H ) is a pro-p i group.Since we know a priori that Q i is a pro-π group, this is achieved by demanding that every element of Q i is a q i -th power, for This can be expressed by an ∀∃-sentence at the level of H , because Z(H ) = k i=1 C i is quantifier-free definable subject to the parameters x s,t .
Step 4. By Step 2, the group G/H is interpretable in G and finite of (π, r )-bounded order.There is an ∃∀∃-sentence that expresses that the factor group G/H is among the finitely many finite groups of rank at most r and exponent dividing f (π, r ); compare with Example 3.1.Using our results from Step 2, Step 3 and Proposition 3.2, we produce an ∃∀∃-sentence that expresses that the power word x f (π,r ) has (π, r )bounded width in G and that H = G f (π,r ) is pronilpotent of rank at most r .
The conjunction of these two sentences yields an ∃∀∃-sentence η π,r such that • every pro-π group G of rank rk(G) = r satisfies η π,r ; • conversely, if a pro-π group G satisfies η π,r , then H = G f (π,r ) ⊴ o G is pronilpotent and both H and G/ H have rank at most r ; in particular, this ensures that rk( G) ≤ R for R = 2r .
Using the same approach as in the proof of Theorem 1.1, we find an ∃-definable and hence quantifier-neutral subgroup H ⊴ o G that is pronilpotent and has (π, r )bounded index in G; moreover the arrangement can be expressed by means of a suitable ∃∀∃-sentence.We put m = m(r ) = ⌈log 2 (r )⌉ + 1.In the proof of Proposition 3.2 we saw that we can use an ∃∀∃-sentence to describe that m (H ) is semipowerful and of (π, r )-bounded index in H ; in parallel we can realize m (H ) as an ∃-definable and hence quantifier-neutral subgroup.The Sylow subgroup dimensions do not change if we pass from G to an open subgroup.Replacing G by m (H ), we may therefore assume without loss of generality that G itself is pronilpotent and semipowerful.
As G is pronilpotent, G is the direct product of its powerful Sylow subgroups; let G p denote the Sylow pro-p subgroup and T p its torsion subgroup.By Theorem 1.4 it suffices to produce an ∃∀∃-sentence which pins down within the finite range Clearly, the closed subset {g ∈ G | g p = 1} ⊆ c G is quantifier-free definable in G.Moreover, its size equals p d(T p ) and is thus at most p r .We can easily identify by means of an ∃∀-sentence its precise size and hence the invariant d(T p ). □ (2-2)|G : H | < |G : H | and d( H ) = r p , which yields the required contradiction.
. ., ãn ∈ M. The p-power map g → g p induces a surjective L-invariant homomorphism α : M → M between finite abelian p-groups.This implies | M| > |M| and thus |G : H | < |G : H |. Furthermore, using the identity map on L in combination with α, we obtain a surjective homomorphism from H = L ⋉ M onto L ⋉ M = H .This shows that r p = d(H ) ≤ d( H ) ≤ r p and hence d( H ) = r p , which completes the proof of (2-2).(ii) Now suppose that rk(G) ≤ R. Clearly, the maximal local rank mlr(G) = max {rk p (G) | p prime} is at most rk(G).Conversely, Lucchini [1997, Theorem 3 and Corollary 4] established that rk(G) ≤ mlr(G) + 1, with equality if and only if there are • an odd prime p such that r p = rk p (G) = mlr(G) and suffices to produce a runaway couple ( H , N ) for the group HA with respect to p such that |HA : H | < |HA : H |; thus we may assume that G = HA.This reduction allows us to conclude that p (N ) ∩ A ⊴ G and there is no harm in assuming p (N ) ∩ A = 1.Likewise M = H ∩ A ⊴ G, and reduction modulo p (N ) induces an embedding of M ≤ N into the elementary abelian group N / p (N ) ∼ = C r p p .Using (2-3), we conclude that

Example 2. 3 .
Let n ∈ ‫ގ‬ and consider the metabelian pro-p group G = C ⋉ A, where C = ⟨c⟩ ∼ = ‫ޚ‬ p , A = ⟨a 1 , . . ., a n ⟩ ∼ = ‫ޚ‬ n p and the action of C on A is given by Thus any subgroup H ≤ o G with H F = HF = ⟨c⟩F and d( H ) = d( H n (F)/ n (F)) requires less than d(H ) = n + 1 generators, but nevertheless rk(G) = rk(G/ (F)).The group K = ⟨c p , a 1 , . . ., a n ⟩, where the right-hand side denotes the set of all products y 1 • • • y k with factors y i ∈ D i for i ∈ {1, . . ., k}; {0, 1, . . ., r } the invariantsd(G p ) = log p |G p : (G p )| and d(T p ) = log p | {1} (G p )|, where {1} (G p ) = {g ∈ G p | g p = 1}is the set of all elements of order 1 or p.We observe thatG p / (G p ) ∼ = G/ p (G) is essentially the p-Frattini quotient of G and that {1} (G p ) = {g ∈ G | g p = 1}.The Frattini quotient G/ (G) has (π, r )-bounded order and maps onto the p-Frattini quotient G/ p (G).As in the proof of Proposition 3.2, the group G/ (G) is interpretable in G.There is an ∃∀∃-sentence which detects any prescribed isomorphism type of G/ (G) among a (π, r )-bounded number of possibilities; compare with Example 3.1.Forming a suitable disjunction, we can also detect the isomorphism type of the p-Frattini quotient G/ p (G) and hence the minimal numbers of generators d(G p ).
, Lemma 1.23].Thus β 1 holds for H if and only if d(H ) ≤ r .Moreover, in this case (H ) = B(H ) is ∃-definable in H and hence quantifierneutral in the sense of Example 3.1.By recursion, there is an ∃∀∃-sentence β m+1 such that β m+1 holds for H if and only if (3-1) rk( j (H )/ j+1 (H )) ≤ r for 0 ≤ j ≤ m; in this case the subgroup F = m (H ) is ∃-definable in H and hence quantifierneutral, moreover it satisfies d(F) ≤ r .Furthermore, there is an ∀∃-sentence γ which expresses that every Sylow subgroup of F is powerful, viz.that F is semipowerful in the terminology introduced in [Nies et al. 2021, Section 5].Indeed, by [Dixon