we give the concept of ‘braiding’ for 2-groupoids, and we show that this structure is equivalent tobraided regular, crossed modules.

the growing demand for enhanced esthetic appearance has led to great developments in bleaching products. the exposure of hard tissues of the tooth to bleaching agents can affect the roughness of the enamel surface. the freshly bleached enamel surface exposed to various surface treatments such as fluoride and other remineralizing agents have been assessed in this study. the aim of this experimen...

Let p be a prime. It is a fundamental problem to classify the absolute Galois groups GF of fields F containing a primitive pth root of unity ξp. In this paper we present several constraints on such GF , using restrictions on the cohomology of index p normal subgroups from [LMS]. In section 1 we classify all maximal p-elementary abelian-by-order p quotients of these GF . In the case p > 2, each ...

In the 1970s, Isaacs conjectured that there should be a logarithmic bound for the length of solvability of a p-group G with respect to the number of different irreducible character degrees of G. So far, there are just a few partial results for this conjecture. In this note, we say that a pro-p group G has property (I) if there is a real number D = D(G) that just depends on G such that for any o...

We study 3-dimensional Poincaré duality pro-p groups in the spirit of work by Robert Bieri and Jonathan Hillmann, show that if such a group G has nontrivial finitely presented subnormal subgroup infinite index, then either is cyclic normal or polycyclic Demushkin an open G. Also, we describe centralizers generated subgroups groups.