On the number of conjugacy classes of finite p-groups ^{∗}

A. Jaikin-Zapirain

Departamento de Matemáticas, Facultad de Ciencias,

Universidad Autónoma de Madrid

October 18, 2002

Abstract

In this work we study the behaviour of the number of conjugacy classes of finite p-groups using pro-p groups. We introduce the conjugacy growth function r_{n}(G) = max{r(G/N) | N ◁_{o} G, |G : N| = n}. We prove that there are no infinite pro-p groups of linear conjugacy growth (i.e. there is no c such that r_{n}(G) ≤ c log_{2} n for all n > 1) and we show that many known pro-p groups are of exponential conjugacy growth (i.e. there exists ϵ > 0 such that r_{n}(G) ≥ n^{ϵ} for infinitely many values of n).

1 Introduction

Let G be a finite p-group and let r(G) denote the number of conjugacy classes of G. Very little is known about how r(G) depends on the order of G. By a result of P. Hall (for example, see [4, Theorem 15.2]) if |G| = p^{m} and m = 2n + e, where e = 0, 1, then there exists a non-negative integer a = a(G), which we call the abundance of G, such that

r(G) = p^{e} + (p^{2} − 1)(n + a(p − 1)).

In particular, this implies that r(G) ≥ \frac{p^{2}−1}{2}(log_{p} |G| − 1).

In [16] L. Pyber posed the following problem:

∗This work has been partially supported by the DGICYT Grant.

Question 1. Is there a constant c = c(p) such that there are infinitely many p-groups G for which r(G) < c log_{p} |G|?

There had been some speculation that finite quotients of the Nottingham group N_{q} = N(𝔽_{q}) would provide an infinite series of finite p-groups with few conjugacy classes and, moreover, they would give an affirmative answer to Pyber’s question. The next theorem, that will be proved in Section 2, disproves this expectation.

Theorem 1.1. Let q be a power of a prime number p . If p > 2, then there exists a constant c > 0 (depending on q) such that for any normal subgroup N of N_{q},

r(N_{q}/N) ≥ c|N_{q}/N|^{1/(3p)}.

If p = 2 then for infinitely many normal subgroups N of N_{q}, r(N_{q}/N) ≥ |N_{q}/N|^{1/6}.

The behaviour of the number of conjugacy classes of finite quotients of the Nottingham groups of small order was studied previously in [14, 18].

Theorem 1.1 has suggested us to study the behaviour of the number of conjugacy classes of the finite quotients of other pro-p groups. We obtain the analogue result for t-linear pro-p groups (linear over some profinite ring, see [7]) and R-analytic pro-p groups (see [2]):

Theorem 1.2. Let G be an infinite t-linear or an R-analytic pro-p group. Then there exist infinitely many normal open subgroups N of G and a constant ϵ > 0, such that r(G/N) ≥ |G/N|^{ϵ}.

We devote Section 3 to the proof of this theorem.

By analogy with the subgroup growth function a_{n}(G) (see the chapter of A. Mann in [13]), we introduce another function r_{n}(G), which we shall call the conjugacy growth function:

r_{n}(G) = max{r(G/N) | N ◁_{o} G, |G : N| = n}.

We will say that G has polynomial conjugacy growth (PCG) if there exists a polynomial f, such that r_{n}(G) ≤ f(log_{p} n). If the function f is linear, then we say that G has linear conjugacy growth. If there exists a constant ϵ > 0, such that r_{n}(G) ≥ n^{ϵ} for infinitely many values of n, we say that G has exponential conjugacy growth. In other words, if we define

ϵ(G) = lim sup_{N◁_{o}G,|G/N|→∞} \frac{log_{p} r(G/N)}{log_{p} |G/N|} = lim sup_{n→∞} \frac{log_{p} r_{n}(G)}{log_{p} n},

then G has exponential conjugacy growth if ϵ(G) > 0.

It is easy to see that any pro-p group has the same conjugacy growth type as its open subgroups. Theorems 1.1 and 1.2 suggest us the following natural question:

Question 2. Is there an infinite finitely generated pro-p group of polynomial conjugacy growth?

Note that if a such group G exists, then there exists a hereditarily just infinite pro-p group with PCG as well. Indeed, the property of having PCG is inherited by quotients and open subgroups. Hence, since every infinite finitely generated pro-p group can be mapped onto a just infinite pro-p group (see, for instance, [9]), we can suppose that G is just infinite. Now, we can apply the dichotomy concerning the structure of just infinite pro-p groups (see Theorem 3 in the chapter of R. Grigorchuk in [13]), which says that either G is a branch pro-p group, or G contains an open normal subgroup which is isomorphic to the direct product of a finite number of copies of some hereditarily just infinite pro-p group H. Now, note that a branch pro-p group G cannot have PCG, because for any positive integer k, G has a normal open subgroup which is isomorphic to the direct product of m > k copies of some pro-p group L. Therefore only the second possibility can occcur. Obviously, H has PCG, and so it is the group we were looking for.

In [13, p. 71] we can find the list of all the known (at the moment) hereditarily just infinite pro-p groups. We see that only Fesenko’s groups are not included in Theorems 1.1 and 1.2. But it is an easy consequence of the structure of Fesenko’s groups that they are not of PCG. Therefore, an affirmative answer to Question 2 would give new examples of hereditarily just infinite pro-p groups.

Actually we do not know of any pro-p group G which is not of exponential conjugacy growth, i.e. with ϵ(G) = 0. In the next theorem we show that at least ϵ(G) is not bounded below by some positive constant depending only on p:

Theorem 1.3. Let D be a central division algebra over ℚ_{p} of dimension n^{2} and G an open normal pro-p subgroup of D^{∗}/ℚ_{p}^{∗}. Then ϵ(G) ≤ 1/(n + 1).

Since D^{∗}/ℚ_{p}^{∗} has a two generated open subgroup (see, for example, [1]), we obtain the following corollary of this result which is a negative answer to a question posed by A. Shalev ([21]):

Corollary 1.4. There is no a constant ϵ = ϵ(p) > 0 such that r(G) ≥ |G|^{ϵ} for any two generated finite p-group G.

Remark. The construction of a p-group G with r(G) < |G|^{ϵ} for given ϵ > 0 is also done in [17, 3]. I would like to thank L. Pyber for drawing my attention to these works.

Now, suppose that G is a finite p-group of class c. From [22] we know that

r(G) ≥ c|G|^{1/c} − c + 1.  (1)

This implies that

ϵ_{p}(c) = inf\{\frac{log_{p} r(G)}{log_{p} |G|}|G is a finite p-group of nilpotency class c\} ≥ \frac{1}{c}.

Our next result shows that actually we have the equality:

Theorem 1.5. We have ϵ_{p}(c) = 1/c.

In Section 5 we show that there exists a relation between a_{n}(G) and r_{n}(G):

Theorem 1.6. Let G be a finitely generated pro-p group of polynomial conjugacy growth. Then there exists a constant c such that a_{n}(G) ≤ n^{c log_{p} n} for all n > 1.

Note that if a pro-p group G is not linear over ℤ_{p}, then, as it was proved by A. Shalev in [19], for any c < 1/8 there are infinitely many values of n such that a_{n}(G) > n^{c log_{p} n}. Hence if G has PCG, then there are c_{1} > 0 and c_{2} ≥ c_{1}, such that n^{c_{1} log_{p} n} ≤ a_{n}(G) ≤ n^{c_{2} log_{p} n} for infinitely many values of n.

In [6] it is proved that there exist only finitely many finite p-groups with given abundance. The next theorem improves that result:

Theorem 1.7. Let G be a finite p-group such that for every proper normal subgroup N of G

r(G/N) < c log_{p} |G/N|.

Then the order of G is (p, c)-bounded (bounded above by some function depending only on p and c).

This theorem gives us two corollaries. The first one is an immediate application to conjugacy growth. In fact, this result is equivalent to Theorem 1.7.

Corollary 1.8. There are no infinite pro-p groups of linear conjugacy growth.

In the next corollary we consider finite nilpotent groups with bounded number of characters of any degree.

Corollary 1.9. There exists a function f(r) which depends on r such that if G is a finite nilpotent group satisfying the condition

|{χ ∈ Irr(G)|χ(1) = n}| ≤ r for every n,

then |G| ≤ f(r).

In our last result we consider an analogue of Corollary 1.9 for conjugacy classes. In this case we can prove a stronger result:

Theorem 1.10. Let A = (a_{i}) be a sequence such that ∑_{i=0}^{∞} \frac{a_{i}}{p^{i}} converges. Then there exists a number b, which depends on A and p such that if a finite p-group G satisfies

|{Cl_{G}(x)||C_{G}(x)| = p^{i}}| ≤ a_{i},

then |G| ≤ b.

In particular, there exists a function f(r) such that if a finite nilpotent group G has at most r conjugacy classes of size k for any k, then |G| ≤ f(r).

The notation is standard. If M is a subset of G, then we denote by r_{G}(M) the number of conjugacy classes that have a non-empty intersection with M.

Acknowledgments. I would like to thank M. Isaacs, A. Mann, A. Moretó and J. Sangroniz for many fruitful conversations.

2 The Nottingham group

In this section we present the proof of Theorem 1.1. Let q be a power of a prime number p and 𝔽_{q} the field with q elements. The Nottingham group over 𝔽_{q}, denoted by N_{q}, is defined to be the group of normalized automorphisms of the ring of formal power series 𝔽_{q}[[t]]:

N_{q} = {ϕ ∈ Aut(𝔽_{q}[[t]]) | tϕ = t(1 + tf), f ∈ 𝔽_{q}[[t]]}.

In this section we fix q and denote the Nottingham group N_{q} by J. We define the following subgroups of J:

J_{n} = {ϕ ∈ Aut(𝔽_{q}[[t]]) | tϕ = t(1 + t^{n}f), f ∈ 𝔽_{q}[[t]]}.

We begin with an auxiliary result:

Lemma 2.1. Let ϕ, ψ ∈ J_{n} and suppose that tϕ = t + k(t) and tψ = t + s(t). Then

t(ψ ∘ ϕ ∘ ψ^{−1} ∘ ϕ^{−1}) ≡ t + k(t)s′(t) − s(t)k′(t) (mod t^{3n+1}),

where s′ and k′ are derivatives of s and k, respectively.

Proof. First we see that the following congruences hold:

t(ψ ∘ ϕ) = t + k(t) + s(t + k(t)) ≡ t + k(t) + s(t) + s′(t)k(t) (mod t^{3n+1}), t(ϕ ∘ ψ) = t + s(t) + k(t + s(t)) ≡ t + s(t) + k(t) + k′(t)s(t) (mod t^{3n+1}).

Hence

t(ψ ∘ ϕ) − t(ϕ ∘ ψ) ≡ s′(t)k(t) − k′(t)s(t) (mod t^{3n+1}).

Now, applying ψ^{−1} ∘ ϕ^{−1} and bearing in mind that ψ^{−1} ∘ ϕ^{−1} ∈ J_{n}, we obtain

t(ψ ∘ ϕ ∘ ψ^{−1} ∘ ϕ^{−1}) ≡ t + (k(t)s′(t) − s(t)k′(t))ψ^{−1} ∘ ϕ^{−1}
≡ t + (k(t)s′(t) − s(t)k′(t)) (mod t^{3n+1}).

Now we are able to prove Theorem 1.1.

Proof of Theorem 1.1. First, we suppose that p > 2. In this case every normal subgroup of J lies between J_{k} and J_{k+2} for some k, and so without loss of generality we can suppose that N = J_{3n} for some n ∈ ℕ.

Let n ≤ k ≤ 2n, ϕ ∈ J_{k} and g ∈ t^{k+1}𝔽_{q}[[t]] such that tϕ = t + g(t). We shall see now that the centralizer of ϕ in J_{n}/J_{3n} is rather large.

Firstly, using the previous lemma, we see that if the action of ψ ∈ J_{n} on t is defined by tψ = t + g(t)h(t^{p}), for some h ∈ 𝔽_{q}[[t]], then ϕ and ψ commute in J_{n}/J_{3n}. Secondly, it is clear that [J_{3n−k}, ϕ] ⊆ J_{3n}. We easily conclude from these two observations that |C_{J_{n}/J_{3n}}(ϕ)| ≥ q^{(n/p)+n}.

Therefore, r(J_{n}/J_{3n}) ≥ q^{n(p+1)/p} and so

r(J/J_{3n}) ≥ r(J_{n}/J_{3n})/|J/J_{n}| ≥ q^{n/p} ≥ |J/J_{3n}|^{1/(3p)}.

If p = 2, then J = N(𝔽_{2^{s}}) does not have finite obliquity and so we cannot suppose that N = J_{3n} as we have done in the case p > 2. However, the argument above shows that r(J/J_{3n}) ≥ |J/J_{3n}|^{1/6} for all n ≥ 1.

3 Linear and analytic groups

In this section we prove Theorem 1.2. Let R be a commutative Noetherian local pro-p ring and 𝔪 its maximal ideal. We always assume that the associated graded ring gr(R) = ⨁_{n=0}^{∞} 𝔪^{n}/𝔪^{n+1} is an integral domain. The concept of an R-analytic group is defined in [2, Chapter 13], where it is shown that every such group contains an open subgroup which is R-standard. The underlying set of an R-standard group G may be “identified” with (𝔪^{l})^{(d)}, where l ∈ ℕ is called the level and d ∈ ℕ_{0} is the dimension of G. The group operation is given by a formal group law, i.e. a d-tuple F = (F_{1}, …, F_{d}) of power series over R in 2d variables, as follows: for all x, y ∈ G = (𝔪^{l})^{(d)} we have

x · y = F(x, y) = (F_{1}(x, y), …, F_{d}(x, y)).

For every n ≥ l, let G_{n} = (𝔪^{n})^{(d)} ⊆ G.

The law F can be written as follows

F(x, y) = x + y + B(x, y) + O′(3),

where B is the sum of all the polynomials in x and y of degree 2 and the expression O′(3) stands for any power series in which every term has total degree at least 3 and has degree at least 1 in each variable. Then we know that B(x, y) is an R-bilinear form and that the commutator of x and y can be expressed as

[x, y] = B(x, y) − B(y, x) + O′(3).  (2)

Our first result is a special case of Theorem 1.2, concerning 𝔽_{p}[[t]]-analytic groups. Note that when we are considering an 𝔽_{p}[[t]]-standard group G we can assume that G is of level 1.

Lemma 3.1. Let G be an infinite 𝔽_{p}[[t]]-analytic pro-p group. Then there exist infinitely many normal open subgroups N of G and a constant ϵ > 0, such that r(G/N) ≥ |G/N|^{ϵ}.

Proof. Without loss of generality we can suppose that G = G_{1} is an 𝔽_{p}[[t]]- standard group. We shall see the inequality claimed in the statement for the subgroups N = G_{3n}. Let d be the dimension of G, k ≥ n and x ∈ G_{k} = (t^{k}𝔽_{p}[[t]])^{(d)}.

Note that, by (2), [h(t)x, x] ∈ G_{3n} for any h ∈ 𝔽_{p}[[t]] and also [G_{3n−k}, x] ≤ G_{3n}. Hence |C_{G_{n}/G_{3n}}(x)| ≥ p^{(d+1)n}. This implies that r(G_{n}/G_{3n}) ≥ p^{(d+1)n} and so

r(G/G_{3n}) ≥ r(G_{n}/G_{3n})/|G/G_{n}| ≥ p^{n} ≥ |G/G_{3n}|^{1/(3d)}.

This proves the result.

We say that a pro-p group is t-linear if it can be embedded as a closed subgroup in GL_{n}(S) for some commutative profinite ring S and some natural number n. In [7] it is shown that if G is a t-linear finitely generated pro-p group, then G is linear over a commutative Noetherian local pro-p ring ([7, Theorem 4.1]).

Lemma 3.2. Let G be an infinite finitely generated t-linear pro-p group. Then G can be mapped onto an infinite ℤ_{p}-analytic or 𝔽_{p}[[t]]-analytic pro-p group.

Proof. Without loss of generality we can suppose that G ≤ GL_{n}(A) is linear over a commutative Noetherian local pro-p ring A. If G is soluble then G can be mapped onto an infinite p-adic analytic group. Therefore, we may assume that G is not soluble. Since the prime radical N(A) of A is nilpotent, the image of G in GL_{n}(A/N(A)) is infinite. Hence we can suppose that A is semiprime.

Let Q be an element of the nth derived subgroup G^{(n)} of G, different from the identity matrix I_{n} and a an entry of the matrix Q − I_{n} different from 0. Consider an ideal J of A that is maximal among the ideals which have emty intersection with {a^{k}, k ∈ ℕ}. Since A is semiprime, we have that J is a prime ideal. Let \overline{G} be the image of G in GL_{n}(A/J). If \overline{G} is finite then \overline{G}^{(n)} = \{1\}, which contradicts the condition a ∉ J. Hence Ḡ is infinite.

The ring Ā = A/J satisfies the following property: every ideal of Ā contains a power of some fixed element (in our case the element ā). Rings with this property are known to have Krull dimension 1 (see [8, Theorem 146]). Hence Ḡ is linear over ℤ_{p} or 𝔽_{p}[[t]]. In the first case we are done. In the second one we have two possibilities, depending on whether Ḡ is soluble or not. On the former case, arguing as before, Ḡ has an infinite p-adic analytic image. On the latter case we can apply Pink’s result [15] and conclude that Ḡ has an infinite 𝔽_{p}[[t]]-analytic image as well.

Now, we can prove Theorem 1.2.

Proof of Theorem 1.2. Obviously, we can assume that G is finitely generated. First suppose that G is t-linear. By the previous lemma, G has an infinite ℤ_{p}- or 𝔽_{p}[[t]]-analytic quotient. In the first case G has finite rank and the theorem follows from [21, Proposition 4.8]. In the second case we finish the proof using Lemma 3.1.

Now, we suppose that G is R-analytic. It is clear that we can assume that G is R-standard, and in this case we can apply [7, Proposition 5.1], that says that G/Z(G) is R-linear.

4 p-Groups of bounded nilpotency class

Proof of Theorem 1.3. It is known that H = D^{∗}/ℚ_{p}^{∗} is a compact p-adic analytic group of dimension n^{2} − 1. Hence, it has an open normal uniform pro-p subgroup P of rank n^{2} − 1. Note that since every open subgroup of H is virtually isomorphic to P, it is enough to prove the statement for P.

A maximal subfield of D has dimension n. We know that there are only finitely many extensions of ℚ_{p} of degree n (see [10, Proposition II.14]). Moreover if K and L are subfields of D and they are isomorphic then, by the Skolem-Noether Theorem, they are conjugated by an element of D^{∗}. Let {K_{i} ⊂ D|1 ≤ i ≤ d} be a set of representatives of the isomorphism classes of maximal subfields of D. If a ∈ D, then ℚ_{p}[a] is a field and so is contained in some maximal subfield of D. Therefore, there exits h ∈ D^{∗}, such that a^{h} ∈ K_{i} for some i.

Now, we want to study the number of H-conjugacy clases in P/P^{p^{s}}. Define S_{i} = (K_{i}^{∗}/ℚ_{p}^{∗}) ∩ P. Then, from the previous paragraph, we can conclude that

r_{H}(P/P^{p^{s}}) ≤ ∑_{i=1}^{d} |S_{i}P^{p^{s}}/P^{p^{s}}|.

But K_{i}^{∗}/ℚ_{p}^{∗} is a p-adic compact group of dimension n − 1, and so S_{i} is an abelian pro-p group of rank n − 1. We obtain that |S_{i}P^{p^{s}}/P^{p^{s}}| ≤ p^{(n−1)s}, whence

r_{H}(P/P^{p^{s}}) ≤ dp^{(n−1)s} = d|P/P^{p^{s}}|^{1/(n+1)},

and so r(P/P^{p^{s}}) ≤ c|P/P^{p^{s}}|^{1/(n+1)}, where c = d|H/P |.

Note that P has finite obliquity (see [9]) so there exists a constant c_{1} such that for every two open normal subgroups N_{1} and N_{2} of P with |P : N_{1}| = |P : N_{2}| we have that |N_{1}/(N_{1} ∩ N_{2})| < c_{1}. In particular, there exists a constant c_{2}, which depends only on G, such that r(P/N_{1}) ≤ c_{2}r(P/N_{2}). Now let N be an open normal subgroup of P and k ∈ ℕ such that |P : P^{p^{k−1}}| ≤ |P : N| ≤ |P : P^{p^{k}}|. Take N′ such that P^{p^{k−1}} ≤ N′ ≤ P^{p^{k}} and |P : N| = |P : N′|. Then we have

r(P/N) ≤ c_{2}r(P/N′) ≤ c_{2}r(P/P^{p^{k}}) ≤ c_{2}c|P/P^{p^{k}}|^{1/(n+1)} ≤ c_{2}cp^{n−1}|P/N|^{1/(n+1)}.

Hence ϵ(P) ≤ 1/(n + 1).

Now we will prove Theorem 1.5. We obtain this theorem as the corollary of Theorem 1.3 and the next proposition.

Proposition 4.1. Suppose that ϵ_{p}(c) > 1/c for some c. Then there exists a constant ϵ = ϵ(p), such that any infinite pro-p group G have infinitely many finite quotients Ḡ with r(Ḡ) ≥ |Ḡ|^{ϵ}.

Proof. It is clear that we can suppose that G/γ_{2}(G) is finite. Hence G/γ_{n}(G) is finite for all n. Let p^{i_{k}} = |G : γ_{(c+1)^{k}}(G)| and put

r_{1} = lim sup(i_{k+1} − i_{k})/i_{k}.

If r_{1} < c, then there exists a number n such that (i_{k+1} − i_{k})/i_{k} < r_{2} < c for all k ≥ n and it implies that i_{k+1} < (r_{2} + 1)i_{k}. By induction, we obtain that i_{n+k} < (r_{2} + 1)^{k}i_{n}. Therefore we see that there exists m such that i_{m} < (c + 1)^{m}, which is impossible because G is not finite. Hence r_{1} ≥ c.

Let 1/ϵ_{p}(c) < r_{3} < c. From the previous paragraph we conclude that there are infinitely many natural numbers k such that (i_{k+1} − i_{k})/i_{k} > r_{3}, that is i_{k} < i_{k+1}/(r_{3} + 1). Since r(γ_{(c+1)^{k}}(G)/γ_{(c+1)^{k+1}}(G)) ≥ p^{(i_{k+1}−i_{k})ϵ_{p}(c)}, we obtain that

r(G/γ_{(c+1)^{k+1}}(G)) ≥ p^{(i_{k+1}−i_{k})ϵ_{p}(c)−i_{k}} ≥
|G/γ_{(c+1)^{k+1}}(G)|^{\frac{(i_{k+1}−i_{k})ϵp(c)−i_{k}}{i_{k+1}}} ≥ |G/γ_{(c+1)^{k+1}}(G)|^{\frac{ϵp(c)r_{3}−1}{r_{3}+1}} .

Note finally that \frac{ϵ_{p}(c)r_{3}−1}{r_{3}+1} > 0 y depends only on p.

5 Relation between the subgroup and conjugacy growths

In this section we prove Theorem 1.6. Let G be a finitely generated pro-p group. We write a_{n}(G) for the number of subgroups of G of index n. We know that a_{n}(G) is finite. We refer the reader to [13] for detailed properties of subgroup growth. For each pro-p group G define

g_{n}(G) = max{d(H) | H ≤_{o} G, |G : H| = n}.

Recall the following result proved in [12].

Lemma 5.1. Let G be a finitely generated pro-p group. If there exists a constant c_{1} such that g_{n}(G) ≤ c_{1} log_{p} n, then there exists a constant c_{2} such that a_{n}(G) ≤ n^{c_{2} log_{p} n}.

Proof of Theorem 1.6. Let f be a polynomial such that r(G/N) ≤ f(log_{p} |G/N|) for every open normal subgroup N of G and denote by k the degree of f.

First we suppose that

lim sup \frac{g_{n}(G)}{log_{p} n} > 3k + 4.

Choose an integer n such that for any m ≥ n and l ≥ 2m we have

f(m + l) < p^{l−m},  (3)
f(m) < m^{k+1}.  (4)

There exists a subgroup H ≤ G of index p^{t} = |G : H| greater than n such that d(H) ≥ (3k + 4)t. The action of G on {Hg} by multiplication on the right induces a homomorphism ϕ: G → Σ_{p^{t}}. Set N = Ker ϕ. The order of the Sylow p-subgroup of Σ_{p^{t}} is less than p^{p^{t}}, so |G : N| < p^{p^{t}}. Let Ḡ = G/Φ(N). Since

f(p^{t} + d(N)) ≥ f(log_{p} |Ḡ|) ≥ r(Ḡ) ≥ p^{d(N)}/p^{p^{t}} = p^{d(N)−p^{t}},

using (3), we obtain that d(N) ≤ 2p^{t}, whence |Ḡ| ≤ p^{3p^{t}}. The relations Φ(N) ≤ Φ(H) and d(H) ≥ (3k + 4)t imply that r(H/Φ(N)) ≥ p^{(3k+4)t} and so, by (4),

r(Ḡ) ≥ r(H/Φ(N))/p^{t} ≥ p^{(3k+3)t} ≥ (log_{p} |Ḡ|)^{k+1} > f(log_{p} |Ḡ|) ≥ r(Ḡ).

We obtain a contradiction. Hence lim sup \frac{g_{n}(G)}{log_{p} n} ≤ 3k + 4 and we conclude that there exists c_{1} such that g_{n}(G) ≤ c_{1} log_{p} n for all n > 1. By the previous lemma, there exists c such that a_{n}(G) ≤ n^{c log_{p} n}.

6 Pro-p groups of linear conjugacy growth

Let G be a finitely generated pro-p group. In order to prove Theorem 1.7 we need some preliminary work.

Lemma 6.1. Let G be a finite p-group and M an elementary abelian normal subgroup of G. Then for any x ∈ G

|C_{G}(x)| ≥ ō(x)|C_{M}(x)|,

where ō(x) denotes the order of the image of x in G/M. Moreover, if ō(x) ≤ t ≤ ln |M| then

|C_{G}(x)| ≥ t|M|^{1/t}.

Proof. Since C_{M}(x) = M ∩ C_{G}(x), |C_{G}(x)/C_{M}(x)| ≥ ō(x). Hence |C_{G}(x)| ≥ ō(x)|C_{M}(x)|.

Now, consider M as a 𝔽_{p}[x]-module. Then M is a direct sum of principal submodules of order ≤ p^{ō(x)}. Hence |C_{M}(x)| ≥ |M|^{1/ō(x)}.

Consider the function f(t) = t|M|^{1/t}. Then f decreases in the interval 1 ≤ t ≤ ln |M|. Hence we have that

|C_{G}(x)| ≥ ō(x)|C_{M}(x)| ≥ ō(x)|M|^{1/ō(x)} ≥ t|M|^{1/t}.

Now we prove Theorem 1.7.

Proof of Theorem 1.7. Define p^{a_{n}} = |G : D_{n+1}|, where D_{n} = D_{n}(G) is the nth dimension subgroup of G. We split the proof in a number of steps.

Step 1. The sequence \{\frac{a_{n}}{n} \} is bounded above by some function α which depends on p y c.

Note that r_{G}(D_{i} \ D_{i+1}) ≥ p^{a_{i}−a_{i−1}} − 1. Hence

r(G/D_{n+1}) ≥ ∑_{i=1}^{n}(p^{a_{i}−a_{i−1}} − 1) ≥ n(\sqrt[n]{p^{a_{n}}} − 1) ≥ \frac{p^{a_{n}/n} − 1}{a_{n}/n}a_{n}.

We obtain that \frac{p^{an/n}−1}{a_{n}/n} ≤ c for every n, an so \frac{a_{n}}{n} is bounded above by some function which depends on p y c.

Now we choose numbers k and n, depending on p and c, which satisfy the following conditions:

1. k ≥ (ln 2)^{−1} and \frac{p^{(t+k)}}{2(2t+k)} ≥ 2cα, for every t ≥ 0.

2. ∑_{i=0}^{n−1} \frac{1}{4i+2k} ≥ 2cα.

Put H = G/D_{2n} and H_{i} = D_{i}/D_{2n}, 1 ≤ i ≤ 2n. We introduce a function ō on H as follows: if x ∈ H then ō(x) is the order of the image of x in H/H_{n}.

Step 2. Let N ≤ H and A = A(N) = \{x ∈ N|ō(x) ≥ \frac{n}{2 log_{p} |H:N|+k}\}. If

|H_{n}| ≥ p^{n}, then |A| ≥ \frac{|N|}{2}.

Suppose that |A| < \frac{|N|}{2}. Define B = \{x ∈ N|ō(x) < \frac{n}{2 log_{p} |H:N|+k}\}. Then

|B| = |N| − |A| ≥ \frac{|N|}{2}.

If x ∈ B then, by Lemma 6.1,

|C_{H}(x)| ≥ ō(x)|H_{n}|^{\frac{1}{ō(x)}} ≥ n\frac{p^{2 log_{p} |H:N|+k}}{2 log_{p} |H : N| + k} .

Since r_{H}(B) ≥ \frac{1}{|H|} ∑_{x∈B} |C_{H}(x)|, we have

r_{H}(B) ≥ n\frac{p^{2 log_{p} |H:N|+k}}{2|H : N|(2 log_{p} |H : N| + k)} ≥

n\frac{p^{log_{p} |H:N|+k}}{2(2 log_{p} |H : N| + k)}.

Now, bearing in mind the choice of k and that log_{p} |H| = a_{2n−1} ≤ 2αn, we obtain that

n\frac{p^{log_{p} |H:N|+k}}{2(2 log_{p} |H : N| + k)} ≥ 2cαn ≥ c log_{p} |H|.

We have obtained a contradiction. Hence |A| ≥ \frac{|N|}{2}.

Step 3. We have |H_{n}| < p^{n}. In particular, D_{2n} is powerful.

We suppose the contrary. By Step 2 we can suppose that for any subgroup N of H, |A(N)| ≥ \frac{|N|}{2}. Now, we have

r(H) = \frac{1}{|H|} ∑_{x∈H} |C_{H}(x)| ≥ \frac{1}{|H|} ∑_{x∈H} ō(x)|C_{H_{n}}(x)|.

Let χ(x) = |C_{H_{n}}(x)| be the permutation character associated with the action of H on H_{n} (see [5, p.68]). Then the last inequality can be rewritten as

r(H) ≥ ⟨ō, χ⟩.

For each 0 ≤ i ≤ n − 1 we fix an element m_{i} ∈ H_{n} in the following way:

First, let 1 ≠ m_{0} ∈ Z(H)∩H_{n}. Now, suppose we have chosen m_{0}, …, m_{k}. Then let m_{k+1} ∈ H_{n}, m_{k+1} ∉ ⟨m_{0}, ⋯, m_{k}⟩ and [H, m_{k+1}] ⊆ ⟨m_{0}, ⋯, m_{k}⟩. It is clear that the elements {m_{i}} lie in different conjugacy classes of H. Put N_{i} = C_{H}(m_{i}). We have |H : N_{i}| ≤ p^{i}.

If S is a set of representatives of the H-conjugacy classes in H_{n}, then we know that χ = ∑_{m∈S} 1_{C_{H}(m)}^{H}. In particular, χ(x) ≥ ∑_{i=0}^{n−1} 1_{N_{i}}^{H}(x) for every x ∈ H.

Note that

⟨ō, 1_{N_{i}}^{H}⟩ = ⟨ō_{N_{i}}, 1_{N_{i}}⟩ ≥ \frac{n|A(N_{i})|}{|N_{i}|(2 log_{p} |H : N_{i}| + k)} ≥ \frac{n}{4 log_{p} |H : N_{i}| + 2k} ≥ \frac{n}{4i + 2k} .

Hence

r(H) ≥ ⟨ō, χ⟩ ≥ ⟨ō, ∑_{i=0}^{n−1} 1_{N_{i}}^{H}⟩ ≥ n ∑_{i=1}^{n−1} \frac{1}{4i + 2k} ≥ 2cαn ≥ c log_{p} |H|.

Again we have a contradiction. Hence |H_{n}| < p^{n}. In particular, we obtain that D_{k} = D_{k+1} for some n ≤ k ≤ 2n − 1. By a result of D. Riley ([2, Theorem 11.5]), D_{2n} es powerful.

Step 4. The rank of G is bounded above by some function which depends on p and c.

Indeed, we have seen that |G/D_{2n}| is bounded by some function depending on p and c. It is easy to see that |D_{2n}/Φ(D_{2n})| is also (p, c)-bounded. Since D_{2n} is powerful, the rank of D_{2n} is (p, c)-bounded. We conclude that the rank of G is (p, c)-bounded.

Step 5. The end of the proof.

By [21, Proposition 4.8], there exists a number l, depending on the rank of G, such that r(G) ≥ |G|^{l}. By Step 4, the rank of G is (p, c)-bounded, whence l is (p, c)-bounded. We have

c log_{p} |G| > r(G) ≥ |G|^{l},

which implies that |G| is (p, c)-bounded.

Proof of Corollary 1.9. First, note that this property holds for any quotient of G, and so for any Sylow subgroup of G. If H ≠ {1} is Sylow p-subgroup of G, then the number of linear characters is at least p. Hence p ≤ r. Moreover, for any proper normal subgroup N of H, we have

r(H/N) ≤ \frac{r}{2} log_{p} |H/N|.

By the previous theorem, we have that |H| is r-bounded. Hence |G| is rbounded.

7 Fixing the number of conjugacy classes of given size

In this section we shall prove Theorem 1.10. The proof is based on the following folklore result (see, for example, [20]).

Lemma 7.1. The rank of finite p-group G is bounded by a function which depends on min_{x∈G} |C_{G}(x)|.

Proof of Theorem 1.10. Fist we shall show that the rank of G is bounded by some function which depends on p and A. Since ∑_{i=0}^{∞} \frac{a_{i}}{p^{i}} converges, there exists s, depending on A and p, such that ∑_{i=s}^{∞} \frac{a_{i}}{p^{i}} < 1. Hence

|\{g ∈ G||C_{g}(g)| ≥ p^{s}\}| ≤ ∑_{i=s}^{∞} \frac{a_{i}|G|}{p^{i}} < |G|.

This implies that there exists an element g ∈ G such that |C_{G}(g)| < p^{s}. By Lemma 7.1, the rank of G is bounded by some number f, which depends on A and p.

From [11], we know that G has a normal powerful subgroup H of index ≤ p^{f(1+log_{2} f)}. Since ∑_{i=0}^{∞} \frac{a_{i}}{p^{i}} converges, there exists t, depending on A and p, such that

∑_{i=t}^{∞} \frac{a_{i}}{p^{i}} < \frac{p − 1}{p^{1+f(1+log_{2} f)}} ≤ \frac{p − 1}{p|G : H|}.

We will show that exp H < p^{t}, and so the order of G is less than p^{f(1+log_{2} f)}p^{ft}.

Suppose that exp H ≥ p^{t}. Let T be the subgroup of H generated by the elements h ∈ H with order at most exp H/p. Since H is powerful, T is proper characteristic subgroup of H.

Define b_{i} = |{Cl_{G}(g) ⊆ H \ T | |C_{G}(g)| = p^{i}}. From the definition of T it follows that b_{i} = 0 for i ≤ log_{p}(exp H) − 1 ≤ t − 1. We have then

|H \ T | = ∑_{i=t}^{∞} \frac{b_{i}|G|}{p^{i}} ≤ ∑_{i=t}^{∞} \frac{a_{i}|G|}{p^{i}} < |H \ T |,

a contradiction. Hence the order of G is bounded by some function, depending on p and A.

Now if G is finite nilpotent group and G has at most r conjugacy classes of size k for any k, then any Sylow p-subgroup has the same property. Note, also, that if H is a Sylow p-subgroup of G and H ≠ {1}, then the number of conjugacy classes of size 1 is at least p. Hence p ≤ r. Since ∑_{i=0}^{∞} \frac{r}{p^{i}} converges, the order of H is r-bounded. Hence |G| is r-bounded.

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