Abstract:
In this paper, we extend the formation $\hat{\mathfrak{J}_{pr}}$,
which is generated by the class $\mathfrak{J}_{pr}$ originally
introduced by Demina and Maslova. The class $\mathfrak{J}_{pr}$
consists of finite groups in which every non-solvable maximal subgroup
has a primary index. Building upon this framework, we introduce and
study two generalized formations, denoted by $\hat{\mathfrak{J}}$ and
$\hat{\mathfrak{J}_{p}}$, which are obtained by involving minimal
non-solvable maximal subgroups and applying a localization approach to
maximal subgroups. We establish new sufficient conditions under which
a finite group belongs to these formations. In addition, we give
examples of non-solvable groups to illustrate the distinctions between
the class $\mathfrak{J}_{pr}$ and its generalizations.
Keywords:formation, non-solvable group, second maximal subgroup, the core of subgroup.
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