Powers of a product of commutators as products of squares

Powers of Commutators as Products of Squares

Products of Commutators and Products of Squares in a Free Group

A classification of the ways in which an element of a free group can be expressed as a product of commutators or as a product of squares is given. This is then applied to some particular classes of elements. Finally, a question about expressing a commutator as a product of squares is addressed.

Quaternionic Product of Circles and Cycles and Octonionic Product for Pairs of Circles

This paper concerns with a product of circles induced by the quaternionic product considered in a projective manner. Several properties of this composition law are derived and on this way we arrive at some special numbers as roots or powers of unit. We extend this product to cycles as oriented circles and to pairs of circles by using the algebra of octonions. Three applications of the given pro...

Product Positioning as a Moderator for Halal Cosmetic Purchase Intention

Consumers have been spoiled by the quality of Halal products available on the market. However, existing literature has lacked recognition for consumer Halal cosmetic products. Therefore, this study enhances the body of knowledge about consumer attitude and intention to purchase Halal cosmetic products. Specifically, this article will discuss the moderating effect positioning has on attitude wit...

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...