Products of Commutators and Products of Squares in a Free Group

Powers of a product of commutators as products of squares

We prove that for any odd integer N and any integer n > 0, the Nth power of a product of n commutators in a nonabelian free group of countable infinite rank can be expressed as a product of squares of 2n+1 elements and, for all such odd N and integers n, there are commutators for which the number 2n+1 of squares is the minimum number such that the Nth power of its product can be written as a pr...

Powers of Commutators as Products of Squares

SOLVABILITY OF FREE PRODUCTS, CAYLEY GRAPHS AND COMPLEXES

In this paper, we verify the solvability of free product of finite cyclic groups with topological methods. We use Cayley graphs and Everitt methods to construct suitable 2-complexes corresponding to the presentations of groups and their commutator subgroups. In particular, using these methods, we prove that the commutator subgroup of $mathbb{Z}_{m}*mathbb{Z}_{n}$ is free of rank $(m-1)(n-1)$ fo...

Commutators and Squares in Free Nilpotent Groups

In a free group no nontrivial commutator is a square. And in the free group F2 = F (x1, x2) freely generated by x1, x2 the commutator [x1, x2], is never the product of two squares in F2, although it is always the product of three squares. Let F2,3 = 〈x1, x2〉 be a free nilpotent group of rank 2 and class 3 freely generated by x1, x2. We prove that in F2,3 = 〈x1, x2〉, it is possible to write cert...

Modification of physicochemical and nutritional properties of gluten free cakes by natural additives application

The demand for gluten-free products has increased in recent years, owing to the market trends and an increase in the number of celiac and other gluten sensitivities and wheat allergies. Products that contain gluten are also a major diet for a large portion of the world's population. For this reason, the development of gluten-free products is not only an urgent need for celiac patients but also ...