A Hossz´u-Gluskin Algebra and a Central Operation of (sm; m)-Groups

Abstract

\begin{abstract}

{In this paper we prove a generalization of the Hossz\'{u}-Gluskin theorem for $(sm,m)$-groups in terms of a $\{1,(s-1)m+1\}$-neutral operation and we define the algebra $\left(Q^{m},\left\{\cdot,\varphi,c_{1}^{m}\right\}\right)$ associated to the $(sm,m)$-group $\left(Q,A\right)$. The central operation of an $(n,m)$-group is defined in \cite{litcop}. Research results of central operation properties using a bijection $\sigma_{\alpha}:Q^{m}\rightarrow Q^{m}$ are presented by means of a series of theorems.  Then, a central operation of an $(sm,m)$-group is investigated using the previously mentioned algebra.}

\end{abstract}