A generalization of the Chermak-Delgado measure on subgroups and its associated lattice (Q6620113)

Let \(G\) be a group acting on itself by conjugation. In [\textit{A. Chermak} and \textit{A. Delgado}, Proc. Am. Math. Soc. 107, No. 4, 907--914 (1989; Zbl 0687.20022)], for every \(r \in\mathbb{R}\), \(r>0\), a measure \(\mu_{r}\) that associates each subgroup \(H \leq G\) with a real number \(\mu_{r}(H)=|H|^{r}|C_{G}(H)|\) was introduced. If \(m_{r}=\max \{ \mu_{r}(H) \mid H \leq G \}\), then the set \(M_{r}(G)=\{ H \leq G \mid \mu_{r}(H)=m_{r} \}\) is a sublattice of the lattice \(\mathrm{Sub}(G)\) of all subgroups of \(G\).\N\NIn the paper under review, the authors consider mappings \(\mathcal{N}: \mathrm{Sub}(G)\rightarrow \mathrm{Sub}(G)\) and define a measure associated to \(\mathcal{N}\) by setting \(\nu(H)=|H||\mathcal{N}(H)|\). They investigate under what conditions on \(\mathcal{N}\) the subgroups with maximal measure form a sublattice of \(\mathrm{Sub}(G)\). In particular, their focus is on the case where \(\mathcal{N}(H)\) is a centralizer-like subgroup.