On Fox and augmentation quotients of semidirect products

On Fox Quotients of Arbitrary Group Algebras

“…is proved to be a direct summand of Q 3 (G, H) but is not computed in [16]; we here fill this gap noting that X = Q H 3 (H, H) where the N-series H = (H (n) ) n≥1 is given by H (n+1) = [H (n) , G], see also [9]. Indeed, the structure of Q …”

“…This map is clearly surjective but rarely globally injective; for instance, θ γ is injective if G has torsionfree lower central quotients G n /G n+1 or is cyclic, but θ γ is non injective for all non cyclic finite abelian groups [1]. At least, the kernel of θ G is torsion as θ G ⊗ Q is an isomorphism; this was proved by Quillen for G = γ and follows from work of Hartley [11] in the general case, see also [9]. Moreover, Ker(θ G ) is trivial in degree 1 and 2 (by [2] for N = γ ) and is explicitely known in degree 3, see [4].…”

“…where ζ G n is an isomorphism by Proposition 2.1. Moreover, θ G here is an isomorphism since G has torsionfree factors [9], whence ν (n−1)1 and θ GH n are isomorphisms, too.…”

“…should be functorially expressable in terms of (generally higher order) operators between suitable polynomial endofunctors of Ab and their derived functors, applied to appropriate abelian subquotients of the nilpotent group in question (here Ab denotes the category of abelian groups). For more examples of this structural phenomenon see also [4], [6], [8], [9], [10]. where a q , b q ∈ G (2) , c r , d r ∈ G, k r ∈ Z Z such that c kr r , d kr r ∈ G (2) for 1 ≤ r ≤ s and g = p q=1 [a q , b q ] s r=1 [c r , d kr r ] ∈ G 3 .…”

Dimensions of Zassenhaus filtration subquotients of some pro-p-groups

“…(a) This follows from [Hart,Theorem 1.3]. In [Hart,Theorem 1.3], a discrete group G is considered, but an adaptation of this proof to the profinite case is straightforward.…”