On linear just infinite pro-p-groups. ^{∗}

A. Jaikin-Zapirain

Departamento de Matemáticas-Matematika Saila Facultad de Ciencias-Zientzi Fakultatea Universidad del País Vasco-Euskal Herriko Unibertsitatea Apdo. 644, 48080-Bilbao, Spain e-mail: mtbjazaa@lg.ehu.es

1 Introduction

Interest on just infinite pro-p groups (i.e. infinite pro-p groups with only finite proper pro-p images) has grown steadily during the last few years, the most ambitious project in this area being the classification of the groups in this family. It is expected that, as in the classification of finite simple groups, except for some ‘sporadic’ groups, any just infinite pro-p group will lie into one of several welldefined families. However, this theory is still in its first stages and no precise conjectures have been posed by the moment. It is therefore natural to add some extra conditions and the one we choose in this work is linearity. We need first to give a precise meaning to a just infinite pro-p group being linear. According to the classical definition, to be linear means of course to be a subgroup of the full linear group of a given degree defined on a field, however this is not entirely appropriate for our purposes since it is does not take any advantage of the topological nature of a pro-p group. The following two definitions, notably the second one, are better suited in our context.

Definition. Let G be a pro-p group. We shall say that G is linear if it is linear over some commutative ring and that G is t-linear if it is a closed subgroup of GL_{n}(A) for some commutative profinite ring A.

Recall that if A is a commutative profinite ring then GL_{n}(A) is a profinite group, and if G is a finitely generated pro-p group and also a subgroup of GL_{n}(A) then G is automatically closed in GL_{n}(A) ([2, Corollary 1.21]). Note that just infinite pro-p groups are finitely generated.

It seems reasonable that for pro-p groups in general t-linearity should be strictly stronger than linearity but no actual contrexamples are known. In [6] it has been conjectured that a non-abelian free pro-p group is not t-linear. By the moment we only know that it is not t-linear of degree 2 (see [13]) and it is not linear over 𝔽_{p}[[t]] (see [1]). Yet, I suspect that this group is linear over some field. In this paper our main interest is focused on the following question posed by C.R. Leedham-Green.

∗This work has been supported by a grant of the Government of the Basque Country and by the DGICYT Grant PB97-0604.

Question. Is every linear just infinite pro-p group linear over ℤ_{p} or 𝔽_{p}[[t]]?

We have not been able to settle this question but we have proved that contrexamples, if they exist, must have some interesting properties. Note that soluble just infinite pro-p groups are linear over ℤ_{p}. Therefore, our attention is devoted to insoluble just infinite pro-p groups. The following result will be very useful for us and is also of certain interest by itself.

Proposition 1.1. Let G be a just infinite pro-p group. Then G is insoluble if and only if every non-trivial normal subgroup of G is open.

So we see that insoluble just infinite pro-p groups are also just infinite as abstract groups. This fact was proved for the Nottingham group by B. Klopsch ([4]). Recall that a pro-p group G is called hereditarily just infinite if every open subgroup of G is just infinite. If G is an insoluble pro-p group then every open subgroup is also insoluble and so Proposition 1.1 implies the following corollary.

Corollary 1.2. Let G be an insoluble hereditarily just infinite pro-p group. Then G is hereditarily just infinite as an abstract group.

If x_{1}, …, x_{n} are elements in a ring we define

s_{n}(x_{1}, …, x_{n}) = ∑_{σ∈S_{n}} sgn(σ)x_{σ(1)} … x_{σ(n)},

where the sum is taken over all permutations of degree n and sgn(σ) is the sign of σ. We say that a ring satisfies the n-th standard identity, or s_{n} for short, if s_{n}(x_{1}, …, x_{n}) = 0 for any elements x_{1}, …, x_{n} in the ring. It is easy to prove that, for a commutative ring A, 𝕄_{n}(A) satisfies s_{n^{2}+1}. Actually, by the Amitsur-Levitzki Theorem, 𝕄_{n}(A) satisfies s_{2n}. If R is a ring we set s_{n}(R) to be the ideal of R generated by {s_{n}(a_{1}, …, a_{n}) | a_{1}, …, a_{n} ∈ R}. Now we give a criterion for a just infinite pro-p group to be linear.

Theorem 1.3. Let G be a just infinite pro-p group. Then G is linear if and only if G∩(1+s_{2n}(ℤ[G])) = {1} for some n. Moreover, in this case, G is linear over some field.

Let Δ be the ideal of ℤ[G] generated by p and \{g − 1 | g ∈ G\} and \overline{s_{2n}(ℤ[G])} the closure of s_{2n}(ℤ[G]) in the Δ-adic topology of ℤ[G]. Hence, \overline{s_{2n}(ℤ[G])} = ∩_{i∈ℕ}(s_{2n}(ℤ[G]) + Δ^{i}).

Theorem 1.4. Let G be a just infinite pro-p group. Then the following conditions are equivalent:

(i) G is linear over ℤ_{p} or 𝔽_{p}[[t]].

(ii) G ∩ (1 + \overline{s_{2n}(ℤ[G])}) = \{1\} for some n.

As a corollary of this theorem we obtain that t-linear just infinite pro-p groups are linear over ℤ_{p} or 𝔽_{p}[[t]]:

Corollary 1.5. Let G be a t-linear just infinite pro-p group. Then G is a closed subgroup of of GL_{m}(ℤ_{p}) or of GL_{m}(𝔽_{p}[[t]]) for some m.

Another proof of this corollary is given in Section 4.

In view of Theorem 1.3 we define the following set 𝕀_{n} of ideals of ℤ[G]: an ideal I belongs to 𝕀_{n} if it is maximal among the ideals J containing s_{2n}(ℤ[G]) and satisfying G ∩ (1 + J) = {1}. We will see that if G is an insoluble linear just infinite pro-p group and I ∈ 𝕀_{n} then ℤ[G]/I is a prime Noetherian PI ring. In the case when G is not t-linear we can say a bit more.

Theorem 1.6. Let G be a just infinite pro-p group. If G is linear but not t-linear (linear over ℤ_{p} or 𝔽_{p}[[t]]), then there exists n ≥ 0 such that for any m ≥ n the family 𝕀_{m} is not empty and for any I ∈ 𝕀_{m}, ℤ[G]/I is a simple ring of finite dimension over its centre.

In [5] the authors consider just infinite pro-p groups which are linear over ℤ_{p} and they prove that these groups are open compact subgroups of the groups of ℚ_{p}-rational points of semisimple algebraic groups. What can we say about just infinite pro-p groups which are linear over 𝔽_{p}[[t]]? The results by Pink in [9] represent a breakthrough in the understanding of these groups. One of the immediate corollaries of this work is an analogue for 𝔽_{p}[[t]]-linear insoluble just infinite pro-p groups: any such group has an open subgroup which is isomorphic to an open compact subgroup of the group of 𝔽_{p}((t))-rational points of a semisimple algebraic group. Here 𝔽_{p}((t)) is the field of quotients of 𝔽_{p}[[t]]. In particular, 𝔽_{p}[[t]]-linear insoluble just infinite pro-p groups are 𝔽_{p}[[t]]-analytic. The converse implication is proved in Section 5. Actually, assuming that R satisfies some natural conditions, we prove:

Theorem 1.7. Let G be an R-analytic just infinite pro-p group. Then G is linear over R (and so it is linear over ℤ_{p} or 𝔽_{p}[[t]]).

In Section 3 we answer in the negative a question raised by A. Shalev in [10, Problem 3] concerning the 𝔽_{p}[[t]]-linearity of the group SL_{n}(ℤ_{p}). Finally, in Section 4 we prove that a finitely generated t-linear pro-p group is linear over some commutative Noetherian local pro-p ring and we conjecture that two ‘minimal’ such rings have the same Krull dimension.

If R is a ring then U(R) is the group of units of R. The derived series of a group is denoted by {G^{(k)}}. The rest of the notation is standard. We refer the reader to [2, 5, 12] for background on pro-p groups and to [7] for background on PI rings. If G is finite p-subgroup of GL_{n}(K), for some field K, then the derived length of G is at most n. We shall use several times these facts, sometimes without mentioning them explicitly.

2 Proofs of the main results

We begin this section with the proof of Proposition 1.1.

Proof of Proposition 1.1. Suppose G is insoluble and let H be a normal subgroup and T its closure in G. Then T is an open subgroup and there exist elements h_{1}, …, h_{d} ∈ H such that T = \overline{⟨h_{1}, …, h_{d}⟩}. Since G is insoluble, [T, T] is open in G. By using the same argument as in the proof of [2, Proposition 1.19], we obtain that [T, T] = {[t_{1}, h_{1}] … [t_{d}, h_{d}] | t_{1}, …, t_{d} ∈ T }, whence [T, T] ≤ H and H is open in G.

Now suppose that G is soluble. Then G has an abelian normal open subgroup A. We pick any non-identity element a in A and take H to be the smallest normal subgroup in G containing a. The ℤ-rank of H is clearly finite, so H cannot be open.

Lemma 2.1. Let R be a ring and G an insoluble just infinite pro-p group contained in U(R). Then for any nilpotent ideal I of R, G ∩ (1 + I) = {1}.

Proof. Suppose G ∩ (1 + I) ≠ {1}. Then by Proposition 1.1, this intersection is an open subgroup of G and since I is nilpotent, it is a nilpotent subgroup. But then G has to be soluble, which is a contradiction.

We shall need the following two results about PI rings.

Proposition 2.2. [7, Theorem 13.6.4] Let R be a semiprime PI ring with centre Z(R) and J a non-trivial ideal of R. Then, J ∩ Z(R) ≠ {0}.

The next proposition is an easy consequence of [7, Theorem 13.4.2].

Proposition 2.3. Let R be a prime PI ring satisfying a polynomial identity of degree 2n. Then there exists an embedding, preserving the identity, R ⊆ 𝕄_{n!}(A), where A is a field.

Now, we are able to prove the next theorem.

Proof of Theorem 1.3. Suppose first that G is linear. Then G is a subgroup of GL_{n}(A) for some commutative ring A. Hence we can construct a ring homomorphism ϕ: ℤ[G] → 𝕄_{n}(A), such that G ∩ (1 + ker ϕ) = {1}. Since s_{2n}(ℤ[G]) ⊆ ker ϕ, the ‘only if’ part of the theorem is proved.

Now suppose G ∩ (1 + s_{2n}(ℤ[G])) = {1} for some n. If G is soluble then G is linear over ℤ_{p} and we are done, so without loss of generality we can suppose that G is not soluble. By using Zorn’s Lemma, we obtain that there exists an ideal I which is maximal with respect to the properties s_{2n}(ℤ[G]) ⊆ I and G ∩ (1 + I) = {1}. Put R = ℤ[G]/I and identify G with a subgroup of the group of units of R. By Lemma 2.1, R contains no non-trivial nilpotent ideals which implies by [7, Corollary 0.2.7] that R is semiprime. Now, let J ≠ {0} be an ideal of R. Since H = G ∩ (1 + J) ≠ {1}, Proposition 1.1 yields that H is an open subgroup of G and so R/J is Noetherian. It is now clear that R satisfies the ascending chain condition on ideals, which enables us apply [7, Theorem 2.15] and conclude that R has only finitely many minimal prime ideals. By Proposition 1.1 we obtain that {0} is the unique minimal prime ideal of R, that is, R is prime. The result follows now from Proposition 2.3.

Remark 2.4. Note also that the ring R in the last proof not only is prime, but also Noetherian by [7, Theorem 13.6.15].

The proof of Theorem 1.4 requires the following lemma.

Lemma 2.5. Let G be a just infinite pro-p group and K a closed ideal of 𝔽_{p}[[G]] such that G ∩ (1 + K) = {1}. Then there exists a closed ideal I containing K maximal with respect to the property G ∩ (1 + I) = {1}.

Proof. All we have to do is to explain why Zorn’s Lemma can be applied here. We take an ascending chain of closed ideals of 𝔽_{p}[[G]] containing K, {I_{k}}, such that G∩(1+I_{k}) = {1} for all indices k and claim that the closure J of the union of the ideals in the chain also satisfies the additional condition G∩(1+J) = {1}. Indeed, otherwise the ring 𝔽_{p}[[G]]/J would be finite, whence J would be finitely generated as, say left, 𝔽_{p}[[G]]-module. We take the unique maximal ideal M of 𝔽_{p}[[G]] and note that J = I_{k} + MJ for some k. But then, by Nakayama’s Lemma, I_{k} = J, which is a contradiction.

Proof of Theorem 1.4. The proof that (i) follows from (ii) is as in the previous theorem.

We assume now that (ii) holds and, without loss of generality, that G is insoluble. Let Λ be the completion of ℤ[G]/\overline{s_{2n}(ℤ[G])} in the Δ/\overline{s_{2n}(ℤ[G])}-adic topology. If Λ̄ = Λ/pΛ is finite then Λ is finitely generated as ℤ_{p}-module and so G is linear over ℤ_{p}. So, we can suppose that Λ̄ is infinite. In this case, G is embedded in U(Λ̄), and by Lemma 2.5, there exists a closed ideal I of Λ̄ satisfying G ∩ (1 + I) = {1} and maximal with this property. Put R = Λ̄/I and let Δ̄ be the unique maximal ideal of R. We regard G as a subgroup of U(R).

The ring R is semiprime because if L is a nilpotent ideal of R, so is its closure, which must be trivial by Lemma 2.1 and the maximality of I. Thus Proposition 2.2 can be applied to pick a non-zero element z in Δ̄ ∩ Z(R). We claim that the annihilator of z is trivial. Actually, any non-trivial closed ideal contains a power of z but no power of z annihilates z itself because R is semiprime, hence the only possibility for the annihilator of z is to be trivial. Now, we set K to be the ideal ∩_{i∈ℕ}z^{i}R. If K ≠ {0} a power of z, say z^{k}, lies in K and then z^{k} = az^{k+1} for some a ∈ R. But this is impossible because the annihilator of z is trivial and z is not invertible. Therefore K = {0}. Now, R/zR is finite so R is a finitely generated free 𝔽_{p}[[z]]-module and this implies that G is linear over 𝔽_{p}[[t]].

Proof of Corollary 1.5. As we will see in Section 4 we can suppose that A is a pro-p ring. The inclusion map G ⊆ 𝕄_{n}(A) can be extended to ℤ[G] and this extension is continuous map if ℤ[G] is endowed with the Δ-adic topology. This implies that \overline{s_{2n}(ℤ[G])} maps to zero, which allows as to appeal to the implication (ii)⇒(i) of Theorem 1.4 to obtain the conclusion desired.

Proof of Theorem 1.6. By way of contradiction, suppose that R = ℤ[G]/I is not simple for some I ∈ 𝕀_{m}. We have seen in the proof of Theorem 1.3, that R is prime. We consider G as a subgroup of U(R). Since R is not simple, there exists a proper non-zero ideal J of R contained in the ideal of R generated by {g − 1 | g ∈ G^{(n!+1)}} and by Proposition 2.2, we can take 0 ≠ z ∈ J ∩ Z(R). The element z enjoys following properties:

(i) For any proper ideal K of R there exists l ∈ ℕ such that z^{l} ∈ K.

(ii) ∩_{i∈ℕ}z^{i}R = {0}.

We prove now these claims.

(i) Let K_{1}/K be the prime radical of R/K. By the construction of R, H = G ∩ (1 + K_{1}) is not trivial and is open in G. Since R/K_{1} is a semiprime PI ring we obtain, by Proposition 2.3, that R/K_{1} can be embedded in ∏_{i} 𝕄_{n!}(A_{i}), where each A_{i} is a field. Hence (G/H)^{(n!+1)} = {1}, that is, G^{(n!+1)} ≤ H. But this means that J ⊆ K_{1} and, in particular, z ∈ K_{1}. Since K_{1}/K is the prime radical of R/K, there exists l ∈ ℕ such that z^{l} ∈ K.

(ii) Since R is prime, the annihilator of z is trivial. Hence, as in the proof of Theorem 1.4, we obtain that ∩_{i∈ℕ}z^{i}R = {0}.

Now we want to show that L = ∩_{i∈ℕ}p^{i}R = {0}. Indeed, if this is not the case, each P_{i} = G ∩ (1 + z^{i}L) is an open subgroup of G and not all of them are equal because their intersection reduces in fact to {1}. So for some j there exists an element g = 1 + z^{j}x ∈ P_{j} \ P_{j+1}, where x ∈ L. Take l such that g^{p^{l}} ∈ P_{j+1}. Then p^{l}x = zy for some y ∈ L. But it is clear from the definition of L that p^{l}L = L and the annihilator of p^{l} is trivial, because R is prime, thus multiplication by p^{l} induces a bijection of L, and the equality p^{l}x = zy implies that x ∈ zL, which is a contradiction because g ∉ P_{j+1}. Hence L = {0}.

We set H = G ∩ (1 + Rz). Now we reach the final contradiction both in the cases pR = {0} and pR ≠ {0}. In the former case, since R/Rz is a quotient of 𝔽_{p}[G/H], it follows that Δ̄^{k} ⊆ Rz for some k ∈ ℕ. But ∩_{i∈ℕ}Rz^{i} = {0}, so I is closed in ℤ[G] with respect to the Δ-adic topology. Hence \overline{s_{2m}(ℤ[G])} ⊆ I and G ∩ (1 + \overline{s_{2m}(ℤ[G])}) = \{1\} which, by Theorem 1.4, means that G is linear over ℤ_{p} or 𝔽_{p}[[t]], against the hypothesis. Finally, if pR ≠ {0}, then Δ̄^{k} ⊆ pR for some k ∈ ℕ and, since ∩_{i∈ℕ}p^{i}R = {0}, I is closed which leads to the same contradiction as before. We conclude that R is simple and, since it is a PI ring, it is of finite dimension over its centre.

3 Linear pro-p groups over fields of characteristic p

Most of results in the previous section can be reformulated considering linearity over rings of characteristic p. This is for instance the analogue of Theorem 1.3 in this new context.

Theorem 3.1. Let G be a just infinite pro-p group. Then G is linear over a ring of characteristic p if and only if G ∩ (1 + s_{2n}(𝔽_{p}[G])) = {1} for some n. Moreover, in this case G is linear over some field of characteristic p.

Recall that SL_{n}^{1}(ℤ_{p}) = Ker{SL_{n}(ℤ_{p}) → SL_{n}(𝔽_{p})}. This is an example of a pro-p group which is linear but not linear over any field of characteristic p.

Theorem 3.2. SL_{n}^{1}(ℤ_{p}) is not linear over any field of positive characteristic.

Proof. Let H be a soluble torsion-free group and suppose it is linear over some field of characteristic q > 0. Then by the Kolchin-Malcev Theorem ([11, Theorem 3.6]), H is virtually triangular, i.e. it has a subgroup of finite index K conjugated to a subgroup of the group of invertible triangular matrices. The commutator subgroup of this group (the unitriangular group) is a q-group so K′ must be trivial. Now note that SL_{n}^{1}(ℤ_{p}) is torsion-free and the group H of triangular matrices inside it is soluble but not virtually abelian.

This last theorem suggests that the following conjecture might be true.

Conjecture 1. Let G be a pro-p group of finite rank. Then G is linear over a field of characteristic p if and only if G is virtually abelian.

At least one part of the conjecture is true: if G is virtually abelian, it is easy to see that it is linear over 𝔽_{p}[[t]]. Moreover, using Pink’s result [9], A. Shalev [8] proved the following particular case of the conjecture.

Theorem 3.3. Let G be a pro-p group of finite rank. If G is linear over 𝔽_{p}[[t]] then G is virtually abelian.

4 Linear dimension of finitely generated t-linear pro-p groups

First of all we prove the result mentioned in the proof of the Corollary 1.5.

Theorem 4.1. Let G be a finitely generated t-linear pro-p group. Then G is linear over some commutative Noetherian local pro-p ring.

Proof. Let A be a profinite ring and G ≤ GL_{n}(A). For each prime number q define the subring A_{q} = {x ∈ A | lim_{n→∞} q^{n}x = 0}. Then A = ∏_{q} A_{q} and GL_{n}(A) = ∏_{q} GL_{n}(A_{q}). Now let G_{q} be the image of G in GL_{n}(A_{q}) under the q-th projection map and J_{q} is intersection of all open maximal ideals of A_{q}. It is easy to see that J_{q} coincides with the Jacobson radical of A_{q}. Now suppose that q ≠ p. If g ∈ 1 + 𝕄_{n}(J_{q}) then lim_{n→∞} g^{q^{n}} = 0 and so G_{q} ∩ (1 + 𝕄_{n}(J_{q})) is trivial. Hence G_{q} can be embedded in ∏_{i} GL_{n}(K_{i}), where K_{i} are finite fields of characteristic q. Let F_{i} be the image of G_{q} in GL_{n}(K_{i}) under the i-th projection map. It is well-known that F_{i} has an abelian subgroup A_{i} of index at most f, which depends only on n . The rank of A_{i} is at most n and so A_{i} has a faithful representation of degree n over ℤ_{p}[w_{i}], where w_{i} is an n_{i}-th root of unity for n_{i} the exponent of group A_{i}. Therefore F_{i} has a faithful representation of degree nf over ℤ_{p}[w_{i}]. Hence G_{q} is linear of degree nf over ∏_{i} ℤ_{p}[w_{i}] and so G ≤ ∏ G_{q} is linear over some pro-p ring.

We are left to deal with the case when A is pro-p ring. Let J be the Jacobson radical of A and put H = G ∩ (1 + 𝕄_{n}(J)). Since G/H can be embedded in ∏_{i} GL_{n}(K_{i}), where K_{i} are finite fields of characteristic p, G/H is finite.

Since H ≤ 1 + 𝕄_{n}(J), H = \overline{⟨1 + b_{1}, …, 1 + b_{s}⟩}, where b_{i} ∈ 𝕄_{n}(J). Let {a_{j} | j = 1, …, k} be the set of elements of A which appear in the matrices b_{i}, R the subring of A generated by {a_{1}, …, a_{k}} and B the closure of R in A. Define a homomorphism ϕ: ℤ[t_{1}, …, t_{k}] → A by means of ϕ(t_{i}) = a_{i}. Let M be the ideal of ℤ[t_{1}, …, t_{k}] generated by p, t_{1}, …, t_{k}. Then ϕ is a continuous map if ℤ[t_{1}, …, t_{k}] has the M-adic topology. Therefore we can extend ϕ to ℤ_{p}[[t_{1}, …, t_{k}]].

We want to prove that ϕ(ℤ_{p}[[t_{1}, …, t_{k}]]) = B and, in particular, B is a Noetherian local pro-p ring. Since ℤ_{p}[[t_{1}, …, t_{k}]] is compact and ϕ is a continuous map, ϕ(ℤ_{p}[[t_{1}, …, t_{k}]]) is closed. Hence

B = \overline{R} = \overline{ϕ(ℤ[t_{1}, …, t_{k}])} ⊆ ϕ(ℤ_{p}[[t_{1}, …, t_{k}]]).

The converse inclusion is obvious.

Now H is of finite index in G, so G is also linear over B.

Let A be a commutative Noetherian local pro-p ring and G a pro-p group. We will say that a homomorphism ϕ: G → GL_{n}(A) is minimal if ϕ is faithful and for every non-trivial ideal I of A the intersection of ϕ(G) and 1 + 𝕄_{n}(I) is different from 1. In this case we write d(ϕ) for the Krull dimension of A. If d(ϕ) is constant for every minimal homomorphism ϕ, we call this constant number the linear dimension of the group G and we denote it simply by dim_{l} G.

Conjecture 2. Every finitely generated t-linear pro-p group has linear dimension.

At this moment we can prove this conjecture only for insoluble t-linear just infinite pro-p groups.

Theorem 4.2. Let G be an insoluble t-linear just infinite pro-p group. Then dim_{l} G = 1.

Proof. Let ϕ: G → GL_{n}(A) be a minimal homomorphism. By Lemma 2.1 A is semiprime. Let Q be an element of ϕ(G^{(n+1)}) different from the identity matrix I_{n} and a an element of the matrix Q − I_{n} different from 0. Consider a maximal ideal J of A which does not intersect {a^{k}, k ∈ ℕ}. It is well-known that J is a prime ideal of A. Let \overline{G} be the image of G in GL_{n}(A/J). If \overline{G} is finite, then \overline{G}^{(n+1)} = \{1\} which contradicts a ∉ J. Hence Ḡ is infinite and so J = \{0\}.

The ring A satisfies the following property: every ideal of A contains a power of some fixed element (in our case the element a). Rings with this property are known to have Krull dimension 1 (see [3, Theorem 146]), so our theorem is proved.

5 Analytic just infinite pro-p groups

Let R be a commutative Noetherian local pro-p ring and 𝔪 its maximal ideal. We 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. To recall what this means, let G be an R-standard group. Then the underlying set of G may be “identified” with the cartesian product (𝔪^{l})^{(d)} of d copies of 𝔪^{l}, for some l ∈ ℕ. The number d ≥ 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_{1}(x, y), …, F_{d}(x, y)).

The neutral element of G is (0, …, 0).

Proposition 5.1. Let G be an R-standard group. Then G/Z(G) is linear over R.

Proof. Let d be the dimension of G. Since G is identified with (𝔪^{l})^{(d)}, A = R[[x_{1}, …, x_{d}]] can be considered as a subring of the ring of functions from G to R. Define on A a structure of R[G]-module by putting (a · y)(x) = a(yxy^{−1}), where a ∈ A and x, y ∈ G. For α = (α_{1}, …, α_{d}), where each α_{i} is a nonnegative integer, put |α| = ∑ α_{i}. Since the group operation in G is given by a formal group law, there exist f_{i,α} ∈ R[[y_{1}, ⋯, y_{d}]] such that

x_{i} · y = x_{i} + ∑_{|α|≥1} f_{i,α}(y)x_{1}^{α_{1}} … x_{d}^{α_{d}}.

Let K be the ideal of R[[y_{1}, …, y_{d}]] generated by W = {f_{i,α}}. Since R[[y_{1}, …, y_{d}]] is Noetherian, there is a finite subset V of W which generates K. Denote by m the maximum of |α| when f_{i,α} ∈ V. Let I = ∑ x_{i}A. Then for any n ∈ ℕ, I^{n} is an R[G]-submodule of A. If y ∈ G acts trivially on I/I^{m+1} then f(y) = 0 for every f ∈ V and so f(y) = 0 for every f ∈ K. Hence x_{i} · y = x_{i} for i = 1, …, d, which means y ∈ Z(G).

Proof of Theorem 1.7. Let H ≤_{o} G be an R-standard group and put N = ∩_{g∈G}H^{g}. It is clear that N is also an open subgroup of G. Since N is normal and closed, so is Z(N). If Z(N) is open then G is soluble, and so p-adic analytic, hence R is finite extension of ℤ_{p} (see for example [2]), whence G is linear over R. Suppose now that Z(N) is trivial. Then N ∩Z(H) = {1} and so by Proposition 5.1, N is linear over R. Since N is of finite index in G, G is also linear over R.

Acknowledgments. I would like to thank C.R. Leedham-Green for suggesting me the study of these groups and for his fruitful conversations. Part of this work was done during my visit to the Queen Mary College and I would like to thank the School of Mathematical Sciences for its hospitality. I also wish to thank J. Sangroniz and D. Segal for some helpful comments on a preliminary version of this work.

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