Let G be a finite group and D division algebra faithfully -graded, dimensional over its center K , where c h r ( ) = 0 . e ∈ denote the identity element suppose ∩ -center of contains ζ n primitive -th root unity, is exponent To such -grading on we associate normal abelian subgroup H positive integer d an o m M μ / Here denotes roots x p Schur multiplier The action trivial induced by Our main th...

Finite nonassociative division algebras (i.e., finite semifields) with 243 elements are completely classified.

The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon thei...

In this article we describe several results based on the paper [Hur98] and which we will refer to as Hurwitz’ theorem. There are several related results: the classification of real normed division algebras, the classification of complex composition algebras and the classification of real composition algebras. The classification of real division algebras is an open problem. Our interest is in co...

A survey of hypercomplex algebras suitable for DSP is presented. Generally applicable properties are obtained, including a paraunitarity condition for hypercomplex lossless systems. Algebras of dimension n = 2 , N ∈ Z, are classified by generation methods, constituting families. Two algebra families, which hold commutative and associative properties for arbitrary N, are examined in more detail:...