The abelianization of a Demushkin group and Labute's q-invariant #
A Demushkin group G of rank n is a one-relator pro-p group, presented on n generators by a
single relator r lying in the Frattini subgroup of the free pro-p group. Abelianizing that
presentation exhibits the topological abelianization as
G^{ab} ≅ ℤ_p^{n-1} × ℤ_p ⧸ q ℤ_p
for a coordinate q of the exponent vector of r (Labute, p. 106). Because r lies in the
Frattini subgroup, q is divisible by p, so the factor ℤ_p ⧸ q ℤ_p is either ℤ_p (when
q = 0) or a finite cyclic group of order p ^ v_p(q) ≥ p. Consequently the torsion subgroup of
G^{ab} is finite and cyclic, and it is trivial exactly when q = 0.
Labute's q-invariant demushkinQ hG is read off from the topological abelianization alone:
it is 0 when G^{ab} is torsion-free and the order of the torsion subgroup otherwise. The
structure theorem is then restated with this invariant as the modulus,
G^{ab} ≅ ℤ_p^{n-1} × ℤ_p ⧸ (q(G)), which shows that q(G) is the invariant q of Labute's
classification, an isomorphism invariant divisible by p. It is the first of the two invariants,
together with the rank, that classify Demushkin groups with q ≠ 2.
Main definitions #
TauCeti.demushkinQ: Labute'sq-invariant of a Demushkin group.
Main results #
TauCeti.IsDemushkin.finite_torsion_topologicalAbelianization,TauCeti.IsDemushkin.isCyclic_torsion_topologicalAbelianization: the torsion subgroup of the abelianization of a Demushkin group is finite and cyclic.TauCeti.IsDemushkin.nonempty_continuousMulEquiv_topologicalAbelianization: the abelianization structure theoremG^{ab} ≃ₜ* ℤ_p^{n-1} × ℤ_p ⧸ (q(G)).TauCeti.IsDemushkin.prime_dvd_demushkinQ:p ∣ q(G), because the relator of a minimal presentation lies in the Frattini subgroup.TauCeti.demushkinQ_eq_zero_iff:q(G) = 0exactly whenG^{ab}is torsion-free.TauCeti.IsDemushkin.exists_demushkinQ_eq_pow: a nonzeroq(G)is a positive power ofp;TauCeti.IsDemushkin.exists_two_le_demushkinQ_eq_pow_of_ne: if moreoverq(G) ≠ p, the exponent is at least2.TauCeti.demushkinQ_congr: theq-invariant is invariant under topological isomorphism.TauCeti.isMulTorsionFree_topologicalAbelianization_of_mulEquiv,TauCeti.demushkinQ_eq_zero_of_mulEquiv,TauCeti.demushkinQ_eq_pow_valuation_of_mulEquiv: the torsion ofG^{ab}read off a modelG^{ab} ≅ ℤ_p^ι × ℤ_p ⧸ (q)withp ∣ q:G^{ab}is torsion-free andq(G) = 0whenq = 0, andq(G) = p^{v_p(q)}otherwise.TauCeti.demushkinQ_presentedProP_eq_zero_iff,TauCeti.demushkinQ_presentedProP_eq_zero_iff_mem_topologicalClosure_commutator,TauCeti.demushkinQ_presentedProP_eq_pow_valuation: for a Demushkin group given by a one-relator presentation⟨X ∣ r⟩withexponentSum r = q • w,w x₀ = 1andp ∣ q, theq-invariant is0exactly whenq = 0, that is whenrlies in the closed commutator subgroup, and isp^{v_p(q)}otherwise.TauCeti.demushkinQ_presentedProP_eq_iff_exists_not_dvd,TauCeti.demushkinQ_presentedProP_eq_iff_exists_degreeOneBasis_repr_inl_ne_zero: for a relatorr ∈ Φ(F)presenting a Demushkin group,q = pexactly when some exponent sum ofris not divisible byp ^ 2, that is, when the class ofringr_1(F)has a nonzerop-power coordinate.
References #
- J. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132, p. 106.
- J.-P. Serre, Galois Cohomology, I §4.5.
Labute's q-invariant of a Demushkin group. It is 0 when the topological abelianization
G^{ab} is torsion-free, which is Labute's q = p^∞ convention, and the number of torsion elements
of G^{ab} otherwise. For a Demushkin group of rank n the torsion subgroup of G^{ab} is finite
and cyclic, and G^{ab} ≅ ℤ_p^{n-1} × ℤ_p ⧸ (q)
(TauCeti.IsDemushkin.nonempty_continuousMulEquiv_topologicalAbelianization).
Equations
Instances For
The q-invariant vanishes when the abelianization is torsion-free.
When the abelianization is not torsion-free, the q-invariant is the number of its torsion
elements.
The q-invariant is an isomorphism invariant.
The q-invariant read off a model of the abelianization #
The model with q = 0 is torsion-free. If G^{ab} ≅ ℤ_p^ι × ℤ_p ⧸ (q) with q = 0, then
G^{ab} is torsion-free.
The q-invariant vanishes on the torsion-free model. If G^{ab} ≅ ℤ_p^ι × ℤ_p ⧸ (q) with
q = 0, then q(G) = 0.
The q-invariant is p^{v_p(q)} on the model with torsion. If
G^{ab} ≅ ℤ_p^ι × ℤ_p ⧸ (q) with q ≠ 0 divisible by p, then the torsion subgroup of G^{ab}
is the cyclic group ℤ_p ⧸ (q) of order p^{v_p(q)}, and that order is q(G).
The torsion subgroup of the abelianization of a Demushkin group is finite: the
abelianization is a topologically finitely generated abelian pro-p group.
The q-invariant vanishes exactly when the abelianization is torsion-free.
The torsion subgroup of the abelianization of a Demushkin group is cyclic.
The q-invariant of a Demushkin group is divisible by p, because the relator of a
minimal presentation lies in the Frattini subgroup.
The abelianization structure theorem for Demushkin groups (Labute, p. 106): for a Demushkin
group G of rank n with q-invariant q,
G^{ab} ≃ₜ* ℤ_p^{n-1} × ℤ_p ⧸ q ℤ_p
as topological groups. When q = 0 the second factor is ℤ_p and G^{ab} ≅ ℤ_p^n is torsion-free;
otherwise it is the cyclic group of order q, which is the torsion subgroup of G^{ab}.
A nonzero q-invariant is a positive power of p: it is the order of the finite cyclic
p-group ℤ_p ⧸ (q), the torsion subgroup of the abelianization.
A q-invariant other than 0 and p is p ^ k with k ≥ 2.
One-relator presentations #
For a Demushkin group given by a one-relator presentation ⟨X ∣ r⟩, the q-invariant is read off
the exponent vector exponentSum r = q • w, w x₀ = 1, of the relator, through the one-relator
abelianization structure theorem TauCeti.presentedProP.oneRelatorAbelianizationEquiv: it is 0
when q = 0, that is when r lies in the closed commutator subgroup, and p^{v_p(q)} otherwise.
The hypothesis p ∣ q is not decoration: the presentation need not be minimal, and a relator with
a unit coordinate, such as x₃ (x₁, x₂) presenting ℤ_p × ℤ_p, has q(G) = 0 while ℤ_p ⧸ (q)
is trivial.
The q-invariant of a one-relator Demushkin group vanishes exactly when the exponent
coordinate q of its relator does, for p ∣ q.
The q-invariant of a one-relator Demushkin group vanishes exactly when the relator lies in
the closed commutator subgroup of the free pro-p group, for p ∣ q.
The q-invariant of a one-relator Demushkin group is p^{v_p(q)} for the exponent
coordinate q ≠ 0 of its relator, p ∣ q: the torsion subgroup of G^{ab} is ℤ_p ⧸ (q).
The q-invariant of a one-relator Demushkin group is p exactly when some exponent sum of
the relator is not divisible by p ^ 2, for a relator all of whose exponent sums are divisible
by p, as they are for a relator in Φ(F). Writing the exponent vector as q • w with a
coordinate w x₀ = 1, the q-invariant is p^{v_p(q)}, and it is p exactly when
v_p(q) = 1.
The q-invariant is p exactly when the relator has a p-power part. For a relator
r ∈ Φ(F) presenting a Demushkin group, the q-invariant of the group is p exactly when the
class of r in gr_1(F) has a nonzero p-power coordinate, that is, when some exponent sum of
r is not divisible by p ^ 2.