Generalized Clifford-Littlewood-Eckmann groups (Q581669)

Given an integer \(n\geq 2\), let G be a group which can be presented as \[ G=<\omega,a_ 1,...,a_ r| \quad \omega^ n=1,\quad a_ i^ n=\omega^{e(i)}\forall i,\quad a_ ia_ j=\omega a_ ja_ i=\omega a_ i=a_ i\omega \forall i>. \] Such groups arise naturally in the study of generalized Clifford algebras and projective representations of finite abelian groups. The study of the structure of such groups when \(n=2\) was undertaken by \textit{T. Y. Lam} and \textit{T. L. Smith} [Rocky Mt. J. Math. 19, 749-786 (1989)]. In this paper, the structure of these groups for arbitrary values of n is determind. The author shows that such a group decomposes in a canonical way as a central product of groups H of order \(n^ 3\), each of which can be presented as \[ H\cong <\omega,a,b| \quad \omega^ n=1,\quad a^ n=b^ n=\omega^ d,\quad ab=\omega ha,\quad \omega \quad central>, \] and if r is odd, one factor group K of order \(n^ 2\) which can be presented as \(K\cong <\omega,c|\) \(\omega^ n=1\), \(c^ n=\omega^ c\), \(\omega c=c\omega >\), where d and e are divisors of n. Furthermore, for all but one of the factors H of G, we have \(d=n\). This decomposition result is useful in studying other properties of these groups, such as their representations and the structure of their associated generalized Clifford algebras. These applications are studied by the author in subsequent papers [Pac. J. Math. (to appear; Zbl 0689.20008)] and [Q. J. Math., Oxf. II. Ser. (to appear)].