21 research outputs found
Mild pro-p-groups with 4 generators
No full textAbstractLet p be an odd prime and S a finite set of primes ≡1 mod p. We give an effective criterion for determining when the Galois group G=GS(p) of the maximal p-extension of Q unramified outside of S is mild when |S|=4 and the cup product H1(G,Z/pZ)⊗H1(G,Z/pZ)→H2(G,Z/pZ) is surjective
Mild pro-2-groups and 2-extensions of Q with restricted ramification
No full textAbstractUsing the mixed Lie algebras of Lazard, we extend the results of the first author on mild groups to the case p=2. In particular, we show that for any finite set S0 of odd rational primes we can find a finite set S of odd rational primes containing S0 such that the Galois group of the maximal 2-extension of Q unramified outside S is mild. We thus produce a projective system of such Galois groups which converge to the maximal pro-2-quotient of the absolute Galois group of Q unramified at 2 and ∞. Our results also allow results of Alexander Schmidt on pro-p-fundamental groups of marked arithmetic curves to be extended to the case p=2 over a global field which is either a function field of characteristic ≠2 or a totally imaginary number field
The lower central series of the group ⟨𝑥,𝑦: 𝑥^{𝑝}=1⟩
No full textIn this paper we determine the Lie algebra associated to the lower central series of the group
⟨
x
,
y
:
x
p
=
1
⟩
\langle x,y:{x^p} = 1\rangle
, p a prime.</p
The determination of the Lie algebra associated to the lower central series of a group
No full textIn this paper we determine the Lie algebra associated to the lower central series of a finitely presented group in the case where the defining relators satisfy certain independence conditions. Other central series, such as the lower
p
p
-central series, are treated as well.</p
Classification of Demushkin Groups
No full textA pro-p-group G is said to be a Demushkin group if(1)dimFp H1(G, Z/pZ) < ∞,(2)dimFp H2(G, Z/pZ) = 1,(3)the cup product H1(G, Z/pZ) × H1(G, Z/pZ) → H2(G, Z/pZ) is a non-degenerate bilinear form. Here FP denotes the field with p elements. If G is a Demushkin group, then G is a finitely generated topological group with n(G) = dim H1(G, Z/pZ) as the minimal number of topological generators; cf. §1.3. Condition (2) means that there is only one relation among a minimal system of generators for G; that is, G is isomorphic to a quotient F/(r), where F is a free pro-p-group of rank n = n(G) and (r) is the closed normal subgroup of F generated by an element r ∈ F9 (F, F); cf. §1.4.</jats:p
On the descending central series of groups with a single defining relation
No full textAbstractIn this paper we show that the Lie algebra associated to the descending central series of a finitely generated group with a single “primitive” defining relation is a Lie algebra with a single defining relation. The proof uses results of [1]
Free Lie algebras as modules over their enveloping algebras
No full textIn this paper we determine the linear relations that exist between the free generators of a free Lie algebra L when it is viewed as a module over its enveloping algebra via the adjoint representation. As an application, the annihilator of a homogeneous element of L is determined.</p
The Lie algebra associated to the lower central series of a link group and Murasugi’s conjecture
No full textThe Chen-Milnor presentation can be used to determine the Lie Algebra associated to the lower central series of the fundamental group of a link in the
3
3
-sphere
S
3
{S^3}
in many interesting cases. We use this fact to obtain new and simpler proofs of unpublished results of Maeda on a conjecture of Murasugi in the sharpened form of Massey and Traldi.</p
The Genesis of a Theorem
No full textIn this article we trace the genesis of a theorem that gives for the first time examples of Galois group of the maximal -extension of , unramified outside a finite set of primes not containing , that are of cohomological dimension . The pro--group is a fab pro--group which means that all its derived factors are finite
