In this note we show that the second homotopy group of B(2, G), classifying space for commutativity a compact Lie G, contains direct summand isomorphic to $$\pi _1(G)\oplus \pi _1([G,G])$$ , where [G, G] is commutator subgroup G. It follows from similar statement E(2, fiber canonical inclusion $$B(2,G)\hookrightarrow BG$$ . As consequence our main result obtain if G) 2-connected, then simply-co...

If p is an odd prime, then the Gupta-Sidki group Gp is an infinite 2-generated p-group. It is defined in a recursive manner as a particular subgroup of the automorphism group of a regular tree of degree p. In this note, we make two observations concerning the irreducible representations of the group algebra K[Gp] with K an algebraically closed field. First, when charK 6= p, we obtain a lower bo...

Abstract Let p be a an odd prime and let G finite -group with cyclic commutator subgroup $G^{\prime }$ G ′ . We prove that the exponent abelianization of centralizer in are determined by group algebra over any field characteristic If, additionally...

Abstract Residual torsion-free nilpotence has proved to be an important property for knot groups with applications bi-orderability and ribbon concordance. Mayland proposed a strategy show that two-bridge group commutator subgroup which is union of ascending chain para-free groups. This paper proves Mayland’s assertion expands the result subgroups link correspond kernels maps $\mathbb{Z}$ . We c...

We link distinct concepts of geometric group theory and homotopy through underlying combinatorics. For a flag simplicial complex $K$, we specify necessary sufficient combinatorial condition for the commutator subgroup $RC_K'$ right-angled Coxeter group, viewed as fundamental real moment-angle $\mathcal{R}_K$, to be one-relator group; Pontryagin algebra $H_*(\Omega \mathcal{Z}_K)$ algebra. also ...

If G is the group of an oriented knot k, then the set Hom(K,Σ) of representations of the commutator subgroup K = [G,G] into any finite group Σ has the structure of a shift of finite type ΦΣ, a special type of dynamical system completely described by a finite directed graph. Invariants of ΦΣ, such as its topological entropy or the number of its periodic points of a given period, determine invari...

For a p-group G and the field F of p elements, let V denote the group of normalized units of the group algebra F G . Generators and relations are provided for V for each group G of order dividing 16. The presentations are sufficiently concise to permit transcription for machine calculation with V . Some applications are described. A theoretical method for obtaining presentations for V is develo...

For each unital $C^*$-algebra $A$, we denote $cel_{CU}(A)=\sup\{cel(u):u\in CU(A)\}$, where $cel(u)$ is the exponential length of $u$ and $CU(A)$ closure commutator subgroup $U_0(A)$. In this paper, prove that $cel_{CU}(A)=2\pi$ provided $A$ an $AH$ algebras with slow dimension growth whose real rank not zero. On other hand, $cel_{CU}(A)\leq 2\pi$ when algebra ideal property no (if further assu...