Lattice Characterizations of \(p\)-Soluble and \(p\)-Supersoluble Finite Groups

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