Constructions of coverings of simple groups using Euclidean lattices

A. K. Martirosov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 25 (2024) no. 1, pp. 161-203

Résumé

The goal of this work is to construct some exceptional (not involved in any infinite family) coverings of finite simple groups. The main tool of this construction is provided by discrete lattices in real and complex Euclidean spaces. Let $G$ be any group. Its covering group is any group $\tilde G$ with a subgroup $Z\leq Z(\tilde G)\cap\tilde G'$ satisfying the condition $\tilde G/Z\cong G$. It is known that, for any finite group $G$, any of its covering groups is also finite, and there is a cover $\tilde G$ of the maximal order. The corresponding subgroup $Z\leq\tilde G$ is uniquely defined up to isomorphism and is called the Schur multiplier of $G$. If $G'=G$ (in particular, if $G$ is non-Abelian simple), then $\tilde G$ is also uniquely defined up to isomorphism. The essence of the method used is as follows. A lattice $X$ considered as an $\mathcal E$-module for some Euclidean ring $\mathcal E$ is constructed in the space $\mathbb R^n$ or $\mathbb C^n$. Since $X$ is discrete, its automorphism group $G$ is finite. The quotient of the lattice $X$ by some of its sublattice $\alpha X$, $\alpha\in\mathcal E$, is considered as a vector space over the field $\mathcal E/\alpha\mathcal E$, on which we define a bilinear or sesquilinear form induced by the usual dot product in $\mathbb R^n$ ($\mathbb C^n$). As a result, the quotient group $G$ by the subgroup of scalar matrices turns out to be embedded in some finite linear group, and the existence of a cover of this linear group or its subgroup now follows. In this work, the method will be demonstrated by the example of exceptional coverings $2\cdot\Omega_8^+(2)$, $3\cdot\mathrm{U}_4(3)$ and $4\cdot M_{22}$. While constructing them, the exceptional covers $3\cdot A_6$, $3\cdot A_7$ (in fact, for $n\ne6,7$, the group $A_n$ does not admit a triple cover) and the embedding $\mathrm{U}_4(3)\mathbin{.}2\hookrightarrow\mathrm{U}_6(2)$ will also be obtained.