Submaximal Soluble Subgroups of Odd Index in Alternating Groups

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {{\mathfrak{X}}} $\end{document} be a class of finite groups containing a group of even order and closed under subgroups, homomorphic images, and extensions. Then each finite group possesses a maximal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {{\mathfrak{X}}} $\end{document}-subgroup of odd index and the study of the subgroups can be reduced to the study of the so-called submaximal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {{\mathfrak{X}}} $\end{document}-subgroups of odd index in simple groups. We prove a theorem that deduces the description of submaximal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {{\mathfrak{X}}} $\end{document}-subgroups of odd index in an alternating group from the description of maximal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {{\mathfrak{X}}} $\end{document}-subgroups of odd index in the corresponding symmetric group. In consequence, we classify the submaximal soluble subgroups of odd index in alternating groups up to conjugacy.