We study the homology of $[P_{n},P_{n}]$, commutator subgroup pure braid group on $n$ strands, and show that $H_l([P_{n},P_{n}])$ contains a free abelian infinite rank for all $1\leq l\leq n-2$. As consequence we determine cohomological dimension $[P_{n},P_{n}]$: $n\geq 2$ have $\mathrm {cd}([P_{n},P_{n}])=n-2$.

Given a group-word w and group G, the set of w-values in G is denoted by Gw verbal subgroup w(G) one generated Gw. The word concise if finite for all groups which finite. We obtain several results supporting conjecture that [u1,…,us] whenever words u1,…,us are non-commutator.

The orthogonal group of a quadratic space over a field has been the focus of intensive research over the past fifty years. As this article will indicate, it continues to be a source of interesting problems. Zassenhaus [17] constructed a decomposition for any element in the orthogonal group of a non-degenerate quadratic space over a field of characteristic not 2. This paper will develop the fund...

Let G be a finite group. We shall designate the commutator subgroup of G by G2= [G, G]; this is the group generated by all commutators [g, h]=ghg~lh~1. Inductively G"=[Gn~~1, G] is defined to be the group generated by commutators of elements of G with elements of GB_1; and G* will designate HiT-i^"It should be recalled that G is nilpotent if G* =P, the subgroup consisting of the identity elemen...

Let G be a finite group. A coprime commutator in is any element that can written as [x,y] for suitable x,y∈G such π(x)∩π(y)=∅. Here π(g) denotes the set of prime divisors order g∈G. An anti-coprime an [x,y], where π(x)=π(y). The main results paper are follows. If |xG|≤n whenever x commutator, then has nilpotent subgroup n-bounded index. every x∈G, H nilpotency class at most 4 [G:H] and |γ4(H)| ...

We explore algebraic characterizations of 2-knots whose associated knot manifolds fibre over lower-dimensional orbifolds, and consider also some issues related to the groups of higherdimensional fibred knots. Nontrivial classical knot groups have cohomological dimension 2, and the knot is fibred if and only if the commutator subgroup is finitely generated, in which case the commutator subgroup ...

We give a complete classification of homomorphisms from the commutator subgroup braid group on n $n$ strands to when is at least 7. In particular, we show that each non-trivial homomorphism extends an automorphism strands. This answers four questions Vladimir Lin. Our main new tool theory totally symmetric sets.

We classify the left-invariant sub-Riemannian structures on unique five-dimensional simply connected two-step nilpotent Lie group with two-dimensional commutator subgroup; this 5D is first twostep beyond three-and Heisenberg groups. Alongside, we also present a classification, up to automorphism, of subspaces associated algebra (together complete set invariants).

We obtain a characterisation of confined subgroups Thompson's group $F$. As result, we deduce that orbital graph point under action $F$ has uniformly subexponential growth if and only this is fixed by the commutator subgroup. This allows us to prove non-embeddability into wobbling groups graphs with growth.