The powerful class of groups

Pro-$p$ groups of finite powerful class are studied. We prove that these are $p$-adic analytic, and further describe their structure when their powerful class is small. It is also shown that there are only finitely many finite $p$-groups of fixed coclass and powerful class.


Introduction
Throughout this paper we assume that p is an odd prime.
Powerful pro-p groups play a fundamental role in Lazard's characterization of p-adic analytic groups [Laz65].In addition to that, their finite counterparts were first systematically discussed by Lubotzky and Mann [LM87], and they turn out to share several properties with abelian groups.
Mann [Man11] introduced the notion of powerful class of a finite p-group G by considering ascending series of normal subgroups with consecutive quotients being powerfully embedded in the corresponding quotient of G.He demonstrated that finite p-groups of small powerful class have well behaved power structure, and thus they are not far away from being powerful.
The purpose of this note is to consider pro-p groups of finite powerful class.These are common generalizations of powerful or nilpotent pro-p groups.Our first main result goes as follows: Theorem.Let G be a finitely generated pro-p group of finite powerful class.Then G is p-adic analytic.The set of all elements of G of finite order forms a finite subgroup of G.
The second part of the above result follows from obtaining a bound for the exponent of Ω i (G) in terms of i and powerful class in the case when G is a finite p-group.The argument relies on techniques developed by Fernández-Alcober, González-Sánchez and Jaikin-Zapirain [FGJ08].
We proceed by looking into pro-p groups of small powerful class.These are closely related to pro-p groups admitting potent filtrations, also known as PF-groups: Theorem.Every finitely generated pro-p group of small powerful class is a PF-group.
González-Sánchez [GS07] showed that torsion-free PF-groups are precisely the p-saturable groups.These groups naturally admit a Lie algebra structure that turns the group into a p-saturable Lie algebra.If G is a finitely generated pro-p group of small powerful class, the above result shows that G is p-saturable.We show that the corresponding Lie algebra also has small powerful class.On the other hand, we exhibit an example showing that Kirillov's orbit method cannot be applied in general to derive the irreducible representations of a torsion-free pro-p group of small powerful class.
If G is a finite p-group of order p n and c is its nilpotency class, then n − c is called the coclass of G. Coclass theory [LGM02] works towards understanding the structure of finite p-groups according to coclass.We show: Theorem.Given p, r and k, there are only finitely many finite p-groups of coclass r and powerful class at most k.
The proof uses Shalev's detailed description of the uniserial structure of large finite p-groups of given coclass, cf.[LGM02].A similar method shows that there are only finitely many PF p-groups of fixed coclass.

Powerful class
Here N p stands for the closure of the abstract group N p , we omit the closure operator throughout the text.It is easy to see that, in the pro-p setting, N is powerfully embedded in G if and only N K/K is powerfully embedded in G/K for all open normal subgroups K of G.If G is powerfully embedded in itself, we say that G is a powerful group.The definition is slightly different when p = 2, as the condition of being powefully embedded is stated as [N, G] ≤ N 4 .But since we always assume that p > 2, we will not use that.
Let G be a finite p-group.Denote by η(G) the largest powefully embedded subgroup of G.Note that η(G) is the product of all powerfully embedded subgroups of G. Clearly we have that Z(G) is contained in η(G).
We recall the notion of powerful class introduced by Mann [Man11].The upper η-series of G is defined by η 0 (G) = 1 and We use the notation pwc(G) = k.Ocasionally we use the shorthand notation It is easily seen that the upper η-series is the fastest growing η-series in a group: Proof.The claim is true for i = 0, 1. Suppose it holds for some i ≥ 1.The group N i+1 /N i is powerfully embedded in G/N i .By induction assumption, we have The notion of powerful class can be extended to the pro-p setting.We say that a pro-p group G has finite powerful class if it has an η-series of closed subgroups of finite length that ends in G. Given a pro-p group G, define η(G) be the product of all closed normal subgroups of G that are powerfully embedded in G. Then η(G) is a closed subgroup of G containing all powerfully embedded subgroups of G.The upper η-series of G can be defined as in the finite case.Then G has finite powerful class if and only there exists k such that η k (G) = G.The smallest such k is the powerful class of G.If a pro-p group G has powerful class ≤ k, then it is an inverse limit of finite p-groups of powerful class ≤ k.
We first collect some properties of the upper η-series.These will be used throughout the text without further reference.
Lemma 2.2.let G be a finitely generated pro-p group.Then Proof.The claim follows from a more general formula Lemma 2.3.Let G be a finitely generated pro-p group.Then the following hold: (1) Proof.Denote Z i = Z i (G).The property (1) obviously holds for i = 0, 1. Suppose the assertion holds for some and the assertion is proved for i+1 as well.In particular, (2) follows directly from here.
(4) is obvious by definition.To prove (5), we use induction on i.We may assume that the inequality holds for i ≥ 1 and for all groups G. Let P = η(η i+1 ).Then we obviously have that η(G) is contained in P .Therefore We proceed by induction.In the case when i = 1, notice that Lemma 2.2 gives The next lemma gives some information on pro-p groups with powerful class two: Lemma 2.4.Let G be a finitely generated pro-p group and suppose that The quotient G/η(G) is thus a powerful group of exponent p, hence it is abelian.

Corollary 2.5. Let G be a finitely generated pro-p group of powerful class
and this concludes the proof.
We are ready to prove the first half of our first main result mentioned in the introduction: Proposition 2.6.Let G be a finitely generated pro-p group of finite powerful class.Then G is p-adic analytic.
Proof.Let pwc(G) = k.We prove the result by induction on k.Clearly, the result holds true for k = 0, 1. Assume it holds for groups of powerful class ≤ k − 1.By Corollary 2.5, we have that It is straightforward to see that if G is a finitely generated pro-p group with an open powerfully embedded subgroup, then G has finite powerful class.When G is nilpotent, the converse also holds: We end this section by mentioning the relationship with capability of groups.We say that a group G is capable if there exists a group Q with Q/Z(Q) ∼ = G.It is well known that non-trivial cyclic groups are not capable.Baer [Bae38] classified finite abelian groups that are capable.We define a finite p-group G to be η-capable if there exists a finite p-group P with P/η(P ) ∼ = G.Again, it is easy to see that a non-trivial cyclic group cannot be η-capable, see, for instance, [DdSMS99,p. 45].Note that if a finite pgroup G is η-capable with P/η(P ) ∼ = G, then (P/η(P ) p )/Z(P/η(P ) p ) ∼ = G by Lemma 2.2.This shows that η-capability implies the usual capability.The converse does not hold.The group C p 2 × C p 2 is capable by [Bae38], yet Lemma 2.4 shows that it is not η-capable, as all abelian η-capable p-groups are elementary abelian.

Pro-p groups of small powerful class
Recall that a pro-p group G is said to have small powerful class if pwc(G) < p.Note that if a stronger condition pwc(G) < p − 1 holds, then G satisfies the condition γ p−1 (G) ≤ G p .Groups satisfying this property are called potent and are thoroughly described by González-Sánchez and Jaikin-Zapirain [GJ04].We are thus more or less only interested in the case pwc(G) = p − 1.
Mann's results on finite p-groups of small powerful class are summarized below.One may verify that similar properties hold for pro-p groups of small powerful class and corresponding closed normal subgroups of small powerful height: Proposition 3.1 ([Man11]).Let G be a finite p-group.
(1) If G has small powerful class, then G p is powerful, and Let G be a pro-p group.A closed normal subgroup Proposition 3.2.Let G be a finitely generated pro-p-group and N a normal subgroup of G.If N has small powerful height, it is PF-embedded in G.
Proof.In the course of the proof, we use Proposition 3.1 (3) without further explicit reference.Let Note that all M i have small powerful height [Man11, Lemma 2.5].Induction shows that we have a descending series . This holds for i = 1, as the fact that we have a central series implies that For induction step, we may assume that Induction on k shows that, under the above assumption, we have Since M p+k = M p k+1 p−1 for all k ≥ 0, we quickly conclude that the intersection of all M i is trivial.This finishes the proof.

Corollary 3.3. Every finitely generated pro-p group of small powerful class is a PF-group.
Every torsion-free pro-p group of small powerful class is therefore p-saturable in the sense of Lazard [Laz65].The latter have a natural Z p -lattice structure, first discovered by Lazard (op.cit.) and further developed by González-Sánchez [GS07].If G is a p-saturable group, then the following operations turn it into a p-saturable Lie algebra G = G: Conversely, every p-saturable Lie algebra becomes a p-saturable group with multiplication given via the Baker-Campbell-Hausdorff formula where u i (x, y) are Lie polynomials in x and y of degree i with coefficients in Q, see [DdSMS99, Theorem 6.28] for further details.
If L is a Z p -Lie algebra, then a subalgebra K is powerfully embedded in L if [K, L] Lie ≤ pK.Analogously, one extends the notion of PF-embedded subgroups to PF-embedded Lie subalgebras [GS07].Furthermore, we can define the powerful class for Z p -Lie algebras as follows.A series 0 If there is an η-series of L that reaches L in finitely many steps, we say that L has finite powerful class.In this case, the length of shortest η-series of L is called the powerful class pwc(L) of L. Denote by η(L) the sum of all powerfully embedded ideals in L. Then we can define the upper η-series of a Lie algebra exactly the same as in the group case.It is also clear that the upper η-series is the fastest growing η-series of the Lie algebra L.

Corollary 3.4. A finitely generated torsion-free pro-p group G has small powerful class if and only if the corresponding Lie algebra G has small powerful class. In this case, pwc(G) = pwc(G) and η
Then all subgroups N i are PF-embedded in G by Proposition 3.2.By [GS07, Theorem 4.5], we have a corresponding series of PF-embedded ideals of G given as 0 The converse follows from the fact that if (N i ) i is an η-series of G, then an analogous argument as in the proof of Proposition 3.2 shows that all N i are PF-embedded in G. Then the argument proceeds along the similar lines as in the previous paragraph.
The equality of the upper η-series of G and G now follows from [GS07, Theorem 4.5].
Kazhdan [Kaz77] showed that Kirillov's orbit method provides a correspondence between the irreducible characters of finite p-groups of class < p and the orbits of the action of that group on the dual space of the corresponding Lie algebra.In [GS09], González-Sánchez showed that the orbit method also works for some classes of p-saturable groups, such as torsion-free potent groups.However, the orbit method no longer works for p-saturable groups of small powerful class: where the action of α on M is given by [x i , α] = x i+1 for i ≤ p − 2, and [x p−1 , α] = x p p and [x p , α] = 1.Then we readily get that where The group G therefore has small powerful class, yet the orbit method does not yield all of its irreducible representations [GS09].
Pro-p whose powerful class is not small may not be PF-groups, as the following example shows: Example 3.6.We exhibit a finite p-group of powerful class equal to p that is not a PF-group.Let M be an elementary abelian p-group with generators x 1 , x 2 , . . ., x p .Form G = α ⋉ M , where α has order p 2 and acts on M as follows: [x i , α] = x i+1 for i = 1, 2, . . ., p − 1, and [x p , α] = 1.The group G has order p p+2 and nilpotency class p.Note that On the other hand, x p is not a p-th power of some element of G.This shows that G is not a PF-group by [FGJ08, Theorem 3.4].By Corollary 3.3 we must have that pwc(G) = p.
On the other hand, there are PF-groups, even torsion-free and potent, which do not have finite powerful class: Example 3.7.In the following we construct a finitely generated torsionfree potent pro-p group G, which does not have finite powerful class.Let p > 3 and let n be a positive integer.Let G n = α ⋉ M , where M = x 1 , x 2 , . . ., x p−2 is an abelian group, and The groups G n clearly form an inverse system.Their inverse limit G ∼ = Z p ⋉ Z p−2 p is topologically generated by two generators, it is torsion-free and potent.As pwc(G n ) are not bounded, the group G does not have finite powerful class.

Elements of finite order and powerful class
In this section we look at the elements of finite order in pro-p groups of finite powerful class k.For i ≥ 0, denote Ω i (G) = x ∈ G | x p i = 1 .At first we bound the exponent of Ω i (G) in terms of i and k: Theorem 4.1.Let G be a finitely generated pro-p group of powerful class k.Suppose k ≤ ℓ(p − 1).Then Ω i (G) p i+ℓ = 1.
Proof.We may assume that G is finite.We prove by induction on ℓ that γ ℓ(p−1) (G) is contained in some PF-embedded subgroup of G.If ℓ = 1, then G has small powerful class, therefore it is a PF-group by Corollary 3.3.For induction step, note that G/η p−1 has powerful class ≤ (ℓ − 1)(p − 1).Therefore there exists a normal subgroup The proof will be concluded once we have shown that [N p i η p−1 , p−1 G] ≤ (N p i+1 η p−1 ) p .To this end, we may assume that (N p i+1 η p−1 ) p = 1 and This proves the claim.We have therefore shown that We note here that Mann [Man11] constucted a finite p-group G of small powerful class with exp Ω 1 (G) > p, therefore the bound given in Theorem 4.1 is close to being sharp.The bound can also be compared with Eeasterfield's bound for the exponent of Ω i (G) in terms of p, i and the nilpotency class of the group G, cf.[FGJ08].
An immediate consequence is the following: Corollary 4.2.Let G be a finitely generated pro-p group of finite powerful class.Then the set of all torsion elements of G forms a finite subgroup of G.

Powerful class and coclass
If G is a finite p-group of class c and order p n , then c < n.The number r = n − c is called the coclass of G. Determining the structure of finite p-groups according to coclass has been very fruitful.We refer to [LGM02].
One of the important features of large p-groups of given coclass is that they act uniserially on certain parts of their lower central series by conjugation.Recall that a finite p-group G acts uniserially on a finite p-group N if |H : [H, G]| = p for every non-trivial G-invariant subgroup H of N .The following result due to Shalev is one of the fundamental results of the coclass theory: holds for all i.Since d > 1 and m i > m i+1 , this is possible only if m i = m i+1 + 1.Then the above η-series of γ m (G) is uniserial, and we thus have that |γ m (G)| ≤ p k .On the other hand, G/γ m (G) has coclass ≤ r and class ≤ m − 1, thus |G : γ m (G)| ≤ p r+m−1 .We conclude that |G| ≤ p k+r+m−1 , and this finishes the proof.

Corollary 5.3. There is no infinite pro-p group of finite coclass and finite powerful class.
We mention here an independent result that can be proved along similar lines: Proposition 5.4.Given p and r, there are only finitely many finite p-groups of coclass r that are PF-groups.
Proof.The proof follows along similar lines like the one of Theorem 5.2.Let G be a PF-group of order p n and coclass r.
Corollary 5.5.There is no infinite pro-p group of finite coclass that is also a PF-group.
A finite p group of order p n and nilpotency class equal to n − 1, where n ≥ 4, is said to be of maximal class.We find here the upper η-series of finite p-groups of maximal class.At first we state the following: Proposition 5.6.Let G be a nonabelian group of order p 3 .
Therefore, if G is a finite p-group of coclass 1, then pwc(G) is equal to the nilpotency class of G. On the other hand, there are several p-groups of coclass two with powerful class strictly smaller than the nilpotency class.For example, there are four powerful p-groups of order p 4 and nilpotency class equal to 2.
, b has small powerful class and a p e = b p e = 1, then

Lemma 5. 1
([LGM02], Theorem 6.3.9).Suppose p > 2. Let G be a finite p-group of coclass r and |G| = p n ≥ p 2p r +r .Let m = p r − p r−1 .Then there exists 0 ≤ s ≤ r − 1 such that G acts uniserially on γ m (G), andγ i (G) p = γ i+d (G)for all i ≥ m, where d = (p − 1)p s .Theorem 5.2.Given p, r and k, there are only finitely many finite p-groups of coclass r and powerful class at most k.Proof.Let G be a finite p-group of coclass r and powerful class k.Denote |G| = p n and suppose without loss of generality that n ≥ 2p r + r.The nilpotency class of G is equal to c = n − r ≥ 2p r .Let m and d be as in Lemma 5.1.We have that pwh γ m (G) ≤ k by [Man11, Lemma 2.5].Consider an η-series 1 Again, assume n ≥ 2p r +r, denote c = n − r ≥ 2p r , and let m and d be as in Lemma 5.1.The group γ m (G) is PF-embedded in G, see [FGJ08, Proposition 3.2].As G acts uniserially on γ m (G), there is a potent filtration of γ m (G) in G that has the form γ