PRO-p GROUPS OF RANK 3 AND THE - Department of ...

PRO-p GROUPS OF RANK 3 AND THE QUESTION OF IWASAWA
ILIR SNOPCE
Abstract. For a prime p > 3 we determine all pro-p groups that satisfy d(G) = d(H) =
3 for all open subgroups H of G.
1. Introduction
Let p be a prime and G a finitely generated pro-p group. It is well known that G is a
free pro-p group if and only if d(H) − 1 = [G : H](d(G) − 1) for every open subgroup H of
G, where d(H) denotes the minimal number of topological generators of H (cf. [5]). For a
positive integer n let En denote the class of all (finitely generated) pro-p groups satisfying
d(H) − n = [G : H](d(G) − n) for all open subgroups H of G. Thus, G belongs to the
class E1 if and only if it is a finitely generated free pro-p group. According to Dummit
and Labute (see [3]) , Iwasawa observed that pro-p groups that belong to the class En
for a fixed positive integer n have interesting representation-theoretic properties and he
raised the question of determining those groups with n = 1. In [3] Dummit and Labute
answered this question for 1-relator groups in the case n = 2. Actually they proved that
G is a Demushkin group if and only if G is a one relator group and G ∈ E2. Note that En
is nonempty for n ≥ 1, because there is a trivial example G = Z n p which obviously belongs
to En. In [8] Yamagishi remarked that no other examples are known when n ≥ 3.
In this note we answer the question of Iwasawa for pro-p groups of finite rank in the case
n = 3 and p > 3. This will provide plenty of nontrivial examples of E3-groups. In fact,
using the recent classification of solvable Zp-Lie lattices of dimension 3 (see [4]), due to
Gonzalez-Sanchez and Klopsch, we will prove the following:
Theorem 1.1. Let p > 3 be a prime. A p-adic analytic pro-p group G belongs to the
class E3 if and only if G is one of the following groups (up to isomorphism):
(1) the abelian group G0, isomorphic to Z 3 p, given with the presentation
G0 = 〈x1, x2, x3 | [x1, x2] = [x1, x3] = [x2, x3] = 1〉;
(2) the non-nilpotent (and meta-abelian) group, parametrised by s ∈ N
G1(s) = 〈x1, x2, x3 | [x2, x3] = 1, [x2, x1] = x ps
2 , [x3, x1] = x ps
3 〉.
2000 Mathematics Subject Classification. 20E18.
1

2 Ilir Snopce
2. Proof of Theorem 1.1
Our first aim is to show that p-adic analytic pro-p groups which belong to the class E3
are torsion free. For that reason we will need the following:
Lemma 2.1. If G is a uniform pro-p group of rank k and p − 1 > k, then G does not
have an automorphism of order p.
Proof. Let G be a uniform pro-p group of rank k. Then G is of dimension k. Corollary
4.18 in [1] indicates that if G is a uniform pro-p group of dimension k, then Aut(G) may
be identified with a subgroup of GLk(Zp). Hence in order to finish the proof of the lemma
it suffices to show that GLk(Qp) has no elements of order p if p − 1 > k, and this is a
consequence of the fact that the p-th cyclotomic polynomial is irreducible over Qp. Indeed,
if A is a matrix of order p acting on a vector space V and v ∈ V , then the annihilator of
v, i.e. the polynomial f of smallest degree such that f(A)v = 0, divides the polynomial
x p − 1. It follows that f is either x − 1 or the p-th cyclotomic polynomial or x p − 1. In
the first case Av = v and in the other cases the subspace spanned by v, Av, A 2 v, ... has
dimension at least p − 1. Thus if A is not the identity then V has dimension at least
p − 1.
We know that a pro-p group G is p-adic analytic if and only if it is of finite rank, i.e. if
and only if there exists a positive constant c such that d(H) ≤ c for every open subgroup
H of G. Now let G ∈ En for some positive integer n be a p-adic analytic pro-p group.
Since d(H) − n = [G : H](d(G) − n) for all open subgroups H of G, and d(H) ≤ c we
must have d(G) = n = d(H). So a p-adic analytic pro-p group G belongs to the class En
if and only if d(G) = n = d(H) for all open subgroups H of G. Now we can prove the
following:
Proposition 2.2. Let k be a positive integer and p a prime such that p − 1 > k. If a
p-adic analytic pro-p group G belongs to the class Ek, then G is torsion free.
Proof. Let G be a p-adic analytic pro-p group that belongs to Ek for p − 1 > k, and let
us assume that G has a torsion element. Since G is p-adic analytic it has a characteristic
uniform pro-p subgroup of finite index, call it U. Let x ∈ G be an element of order p
and H = 〈U, x〉 ≤ G. Then [H : U] = p and U H. Hence H is a semidirect product
of U and 〈x〉. By the previous lemma, Aut(U) does not have an element of order p, so it
follows that H = U × 〈x〉. But in this case we have d(H) = k + 1 since d(U) = k, which
contradicts the fact that G belongs to the class Ek. Hence G does not contain any torsion
element.
Corollary 2.3. Let p > 3 be a prime. Every p-adic analytic pro-p group that belongs to
the class E3 is torsion free.
Now let us recall how we define the Qp-Lie algebra of a p-adic analytic group (see [1] for
more details.). Let G be a p-adic analytic group. Then it has a characteristic subgroup
which is uniform. For every open uniform subgroup H ≤ G, Qp[H] can be made into a

Pro-p groups of rank 3 and the question of Iwasawa 3
normed Qp-algebra, A, and log(H), considered as a subset of the completion  of A, has
the structure of a Lie algebra over Zp. There is a different construction of an intrinsic Lie
algebra over Zp for uniform groups. The uniform group U and its Lie algebra over Zp,
call it LU, are identified as sets and the Lie operations are defined by
g + h = lim (g
n→∞ pn
h pn
) p−n
, [g, h] = lim (g
n→∞ pn
, h pn
) p−2n
= lim (g
n→∞ −pn
h −pn
g pn
h pn
) p−2n
.
This construction also works for saturable groups. It turns out that the Zp-Lie algebra
log(H) is isomorphic to the Zp-Lie algebra LH. In [4] Gonzalez-Sanchez and Klopsch
showed that every torsion-free p-adic analytic pro-p group of dimension less than p is saturable,
and there is a one-to-one correspondence between the p-adic analytic pro-p groups
of dimension less than p and residually-nilpotent Zp-Lie lattices of dimension less than
p. Under this correspondence the closed subgroups correspond to the Zp-Lie sublattices
and the normal subgroups correspond to the Lie ideals. Moreover, the solvable groups
correspond to the solvable Lie lattices.
If H1 and H2 are both open uniform subgroups of G, then H = H1 ∩H2 has finite index
in both H1 and H2, so LH has finite index in both LHi , for i = 1, 2. Hence
Therefore we can unambiguously define
Qp ⊗Zp LH1 = Qp ⊗Zp LH = Qp ⊗Zp LH2.
L(G) = Qp ⊗Zp LH,
where H is any uniform open subgroup of G. Thus L(G) is a Lie algebra over Qp,
of dimension equal to dim(H) = dim(G). From the above discussion we have that
L(G) = Qp ⊗Zp LH ∼ = Qp ⊗Zp log(H). Also let us mention that for a uniform (saturable)
pro-p group U the group structure can be reconstructed from the Lie algebra
stucture of LU by the Baker-Campbell-Hausdorf series.
Before we prove the theorem, we need to recall some notions (see [6]). Let G be a
group and S(G) the set of all finite index subgroups of G. If we define H1 ≥ H2 when
H1 ⊆ H2, then S(G) becomes a directed system. If f is a real-valued function with
domain S(G) then {f(H)}H∈S(G) is a net, so we can talk about lim sup and lim inf.
Denote L f(G) = lim inf{f(H)}H∈S(G) and ¯ Lf(G) = lim sup{f(H)}H∈S(G). Now let G be
a finitely generated pro-p group and f(H) = d(H) =the minimal number of topological
generators of H. In this case S(G) is the set of open subgroups. Since a finite index
subgroup of a finitely generated group is also finitely generated we have d : S(G) → N.
In ([6], Proposition 4.1) Lubotzky and Mann showed that if G is an analytic pro-p
group then L d(G) = d(L(G)), where L(G) is the Lie algebra of G and d(L(G)) denotes
its minimal number of generators. As a consequence of this result they proved that if
L d(G) = ¯ Ld(G) < ∞ for a finitely generated pro-p group G, then G is meta-abelian by

4 Ilir Snopce
finite.
We can apply these results in our case. We observed that if a p-adic analytic pro-p
group G belongs to the class E3 then d(G) = 3 = d(H) for all open subgroups H of G.
Therefore for these groups we have L d(G) = ¯ Ld(G) = 3. Hence d(L(G)) = 3 and G is a
meta-abelian by finite group. In particular G is solvable.
Proof of theorem 1.1. Let p > 3 be a prime and let G be a p-adic analytic pro-p group
that belongs to the class E3. By Proposition 1.2 we have that G is torsion-free and by the
above discussion G is saturable and solvable. Therefore we can associate a 3-dimensional
Zp-Lie algebra LG to G. The Lie algebra LG is solvable and residually-nilpotent. Now let
us list the solvable 3-dimensional and residually-nilpotent Lie algebras for p > 3. All of
them are additively isomorphic to Z 3 p, so they are of the form Zpe1 ⊕ Zpe2 ⊕ Zpe3. Up to
isomorphism the following Lie algebras appear (cf. Proposition 7.3, Theorem 7.4 in [4]):
(1) The abelian algebra,
L0(∞) = 〈e1, e2, e3 | [e1, e2] = [e1, e3] = [e2, e3] = 0〉.
(2) The two-step nilpotent Lie algebras, parametrized by s ∈ N0,
L0(s) = 〈e1, e2, e3 | [e1, e2] = p s e3, [e1, e3] = [e2, e3] = 0〉.
(3) A family of non-nilpotent Lie algebras, parametrized by s ∈ N,
L1(s) = 〈e1, e2, e3 | [e2, e3] = 0, [e2, e1] = p s e2, [e3, e1] = p s e3〉.
(4) A family of non-nilpotent Lie algebras, parametrized by s, r ∈ N and d ∈ Zp,
L2(s, r, d) = 〈e1, e2, e3 | [e2, e3] = 0, [e2, e1] = p s e2 + p s+r de3, [e3, e1] = p s+r e2 + p s e3〉.
(5) A family of non-nilpotent Lie algebras, parametrized by s, r ∈ N0 and d ∈ Zp such
that (a) s ≥ 1 or (b) r ≥ 1 and d ∈ pZp,
L3(s, r, d) = 〈e1, e2, e3 | [e2, e3] = 0, [e2, e1] = p s de3, [e3, e1] = p s e2 + p s+r e3〉.
(6) A family of non-nilpotent Lie-algebras, parametrised by s, r ∈ N0 with s + r ≥ 1,
L4(s, r) = 〈e1, e2, e3 | [e2, e3] = 0, [e2, e1] = p s+r e3, [e3, e1] = p s e2〉.
(7) A family of non-nilpotent Lie algebras, parametrised by s, r ∈ N0 with s + r ≥ 1,
L5(s, r) = 〈e1, e2, e3 | [e2, e3] = 0, [e2, e1] = p s+r ρe3, [e3, e1] = p s e2〉,
where ρ ∈ Z ∗ p is a non-square modulo p.
Assume LG is one of the Lie algebras which belongs to (2), (4), (5), (6) or (7). Then
easily we can see that d(L(G)) ≤ 2 in this case, which contradicts the result of Lubotzky
and Mann that d(L(G)) = 3. So LG belongs to (1) or (3). The case (1) obviously works,
because then G ∼ = Z 3 p. So just the case (3) remains to be checked. If LG is one of the
Lie algebras in the case (3), then G has a presentation as in (2) in the statement of the
theorem. To finish the proof of the theorem it suffices to show that in this case LG is

Pro-p groups of rank 3 and the question of Iwasawa 5
three generated and that every subalgebra of LG of finite index is three generated.
Let K be an additive subgroup of LG of finite index. Since Zp is a Euclidean domain,
there is a ‘triangular’ basis for K:
k1 = αe1 + βe2 + γe3, k2 = µe2 + νe3, k3 = ηe3.
The subgroup K is a subalgebra of LG if and only if it is closed under the Lie bracket.
We have
[k2, k3] = 0, [k3, k1] = [ηe3, αe1 + βe2 + γe3] = αηp s e3 = αp s k3,
[k2, k1] = [µe2 + νe3, αe1 + βe2 + γe3] = αµp s e2 + ανp s e3 = αp s k2.
Hence every additive subgroup of LG is a subalgebra. Moreover, they are Lie algebras of
the same type as LG. From the above calculations of the Lie bracket we also get that
every subalgebra of LG can not be generated with two elements, hence they are three
generated Zp-Lie algebras.
3. Some remarks
We finish the paper by making some general remarks about pro-p groups that belong
to the class En.
Proposition 3.1. Let G be a finitely presented pro-p group and assume that one of the
following holds:
(1) there do not exist a closed subgroup K of G and a finitely generated (as a discrete
group) subgroup X of G normalizing K such that XK/K is an infinite torsion
group;
(2) G does not contain a free non-abelian pro-p group of rank ≥ 2.
If G ∈ En for some positive integer n, then d(G) = d(H) = n for all open subgroups H of
G.
Proof. Let G be a finitely presented pro-p group that satisfies condition (1) in the proposition.
On the one hand, by Wilson’s result ([7], Corollary A) we have that there is a
constant k > 0 such that d(H) ≤ k|G : H| 1
2 for each open subgroup H of G. On the other
hand, since G ∈ En we have that d(H) − n = [G : H](d(G) − n). Hence
n + [G : H](d(G) − n) = d(H) ≤ k|G : H| 1
2
for all open subgroups H of G, which implies that d(G) − n = 0, and consequently
d(H) = n.
Now let G be a finitely presented pro-p group that satisfies condition (2) in the proposi-
tion. By the result of Zelmanov ([2], 7, Theorem A), similarly as in the proof of Corollary
A in [7], we get that there is a constant k > 0 such that d(H) ≤ k|G : H| 1
2 for each open

6 Ilir Snopce
subgroup H of G. Thus, as above we have that d(G) = d(H) = n for all open subgroups
H of G.
Since finitely generated discrete solvable torsion groups are finite, we get the following:
Corollary 3.2. Let G be a finitely presented solvable pro-p group. If G ∈ En for some
positive integer n, then d(G) = d(H) = n for all open subgroups H of G. In particular,
G is p-adic analytic, so if n = 3 and p > 3, then G is one of the groups in Theorem 1.1.
Remark Note that the property that there exists a constant k > 0 such that d(H) ≤
k|G : H| 1
2 for all open subgroups H of G is clearly inherited by homomorphic images.
Therefore the above corollary holds not only for finitely presented solvable pro-p groups,
but also for homomorphic images of such groups.
It seems a very interesting question to investigate E3-groups which are not p-adic analytic.
Acknowledgement I am extremely grateful to my advisor Prof. Marcin Mazur for
his guidance during the preparation of this paper.
References
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Department of Mathematical Sciences, Binghamton University, Binghamton, New York
13902-6000
E-mail address: snopce@math.binghamton.edu