Isomorphism Problem for Almost Simple Linear Groups

The aim of this paper is to contribute to the Isomorphism Problem of complex group algebras which, informally, asks how much the complex group algebra of a finite group G over C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}$$\end{document} know about the structure of the group. In this paper, we show that finite groups G, where PSLn(q)≤G≤PGLn(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {PSL}_n(q) \le G \le \mathrm{PGL}_n(q)$$\end{document}, are uniquely determined (up to isomorphism) by the structure of their complex group algebras. This completes the studies initiated by Shirjian and Iranmanesh (Commun Algebra 46(2):552–573, 2018) and extends the main result of Bessenrodt et al. (Algebra Number Theory 9(3):601–628, 2015) to the family of almost simple linear groups of arbitrary large rank.