Subgroup normality degrees of finite groups I

In this paper, we introduce the probability that a subgroup H of a finite group G can be normal in G, the subgroup normality degree of H in G, as the ratio of the number of all pairs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(h, g)\in H\times G}$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h^g\in H}$$\end{document} by |H||G|. We give some upper and lower bounds for this probability and obtain the upper bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{8}{15}}$$\end{document} for nontrivial subgroups of finite simple groups. In addition, we obtain explicit formulas for subgroup normality degrees of some particular classes of finite groups.