Motivated by the recently introduced concept of a pseudosymmetric braided monoidal category, we define the pseudosymmetric group PSn, as the quotient of the braid group Bn by the relations σiσ −1 i+1 σi = σi+1σ −1 i σi+1, with 1 ≤ i ≤ n − 2. It turns out that PSn is isomorphic to the quotient of Bn by the commutator subgroup [Pn, Pn] of the pure braid group Pn (which amounts to saying that [Pn,...

1. Let G be a connected locally compact group and let G' denote the closure of the commutator subgroup of G. G' is called the derived subgroup of G. Consider the derived series of G, that is, the sequence of subgroups Go, Gi, ■ ■ ■ , Gn, ■ ■ ■ where G0 = G and Gn+i = G'nEach Gn is connected and this sequence becomes stationary at some finite stage,2 that is, for some n, Gn = Gn+i. We define Gn ...

In this paper we will prove analogues of B. Maier’s characterization of weak and strong embeddability of amalgams in N2 [12, 13] for the subvarieties of N2. We will also give analogues of D. Saracino’s characterization of weak and strong amalgamation bases [17], the author’s work on dominions [11] and on amalgamation bases in some varieties of nil-2 groups [8, 9]. Definitions will be recalled b...

the present note arises from the author's talk at the conference ``ischia group theory 2014''. for subgroups $fle n$ of a group $g$ denote by $lat(f,n)$ the set of all subgroups of $n$, containing $f$. let $d$ be a subgroup of $g$. in this note we study the lattice $ll=lat(d,g)$ and the lattice $ll'$ of subgroups of $g$, normalized by $d$. we say that $ll$ satisfies sandwich classification theo...

We study the trace set of commutator subgroup Γ ( 2 ) , a type Local-Global problem about thin groups. determine local obstructions and then use correspondence between binary quadratic forms hyperbolic matrices to find some global obstructions. develop probabilistic argument for existence sufficiently large admissible traces by modeling elements as non-backtracking random walks in 2-dimensional...

For a finite group G= 〈X〉 (X = G), the least positive integer MLX(G) is called the maximum length of G with respect to the generating set X if every element of G may be represented as a product of at most MLX(G) elements of X . The maximum length of G, denoted by ML(G), is defined to be the minimum of {MLX(G) | G = 〈X〉, X = G, X = G−{1G}}. The well-known commutator length of a group G, denoted ...

Let F be a field of characteristic different from 2. It is shown that the problems of classifying (i) local commutative associative algebras over F with zero cube radical, (ii) Lie algebras over F with central commutator subalgebra of dimension 3, and (iii) finite p-groups of exponent p with central commutator subgroup of order p3 are hopeless since each of them contains • the problem of classi...

For a p-group G admitting an automorphism φ of order pn with exactly pm fixed points such that φp n−1 has exactly pk fixed points, we prove that G has a fully-invariant subgroup of m-bounded nilpotency class with (p, n,m, k)-bounded index in G. We also establish its analogue for Lie p-rings. The proofs make use of the theory of commutator-type operators.