Groups of generalized Moufang type and $\mathbb Z_2$-graded algebras

Abstract

A pair $(G,T)$ is called a faithful odd transposition group if $T$ is a normal set of involutions generating the group $G$ and the product of any two distinct elements of $T$ has odd order. We introduce a special subclass of such groups, a \emph{generalized Moufang group of $p$-type} (or $GM(p)$-type), in which the product of any two distinct involutions from $T$ has a fixed prime order $p$. For any such group $(G,T)$ and a scalar parameter $\eta$ in a field $\mathbb F$, we construct a non-associative, non-commutative algebra $A = A_{\mathbb F}(G,T,\eta)$. We prove that every element of $T$ considered as an element of the algebra $A$, is a primitive semisimple idempotent, defining a $\mathbb Z_{2}$-grading of $A$. The Miyamoto group of $A$ with respect to $T$ is isomorphic to $G/Z(G)$. The algebra $A$ contains no nontrivial right ideals and, for a specific choice of the parameter $\eta$, admits a symmetric left Frobenius form. When $G$ is a free Burnside group of odd prime period $p$ extended by an involutory automorphism, the finiteness of $G$ is equivalent to the finite-dimensionality of $A_{\mathbb F}(G,T,\eta)$, providing a reformulation of the Burnside problem. For $p=5$ and $\eta=-1/3$, the algebra generated by two idempotents from $T$ is left-axial and satisfies the Monster-type fusion law $\mathcal{M}(4/3, -4/3)$. For a prime $p>5$, the two-generated algebra is also axial, but obeys a more general fusion law. Although the algebra $A_{\mathbb F}(G,T,\eta)$ is initially defined using a group $GM(p)$-type, we show that it admits an intrinsic, group-free characterization by axiomatizing a class of so-called $GM(p,\eta)$-type algebras. We prove that every algebra in this class is isomorphic to one arising from the construction above, establishing the equivalence of the two definitions.