Lattice Characterizations of $$p$$-Soluble and $$p$$-Supersoluble Finite Groups

Let $$G$$ be a finite group, and let $${\mathcal{L}}(G)$$ be the lattice of all subgroups of $$G$$ . A subgroup $$M$$ of $$G$$ is called modular in $$G$$ if $$M$$ is a modular element (in the Kurosh sense) of the lattice $${\mathcal{L}}(G)$$ , i.e., if (1) $$\langle X,M\cap Z\rangle=\langle X,M\rangle\cap Z$$ for all $$X\leq G,Z\leq G$$ such that $$X\leq Z$$ , and (2) $$\langle M,Y\cap Z\rangle=\langle M,Y\rangle\cap Z$$ for all $$Y\leq G,Z\leq G$$ such that $$M\leq Z$$ . If $$A$$ is a subgroup of $$G$$ , then $$A_{mG}$$ is the subgroup of $$A$$ generated by all its subgroups that are modular in $$G$$ . We say that a subgroup $$A$$ is $$N$$ -modular in $$G$$ ( $$N\leq G$$ ) if, for some modular subgroup $$T$$ of $$G$$ containing $$A$$ , $$N$$ avoids the pair $$(T,A_{mG})$$ , i.e., $$N\cap T=N\cap A_{mG}$$ . Using these notions, we give new characterizations of $$p$$ -soluble and $$p$$ -supersoluble finite groups.