characteristic subgroup

We generalize the Chermak--Delgado measure of a subgroup of a finite group $G$, $\mu(H) = |H||C_{G}(H)|$, and its associated lattice of subgroups with maximal measure. We consider mappings $M$ of the lattice of all subgroups $\mathrm{Sub}(G)$ into itself and define a measure associated to $M$ by setting $\mu(H)=|H||M(H)|$. We investigate under what conditions on $M$ the subgroups with maximal measure form a sublattice of $\mathrm{Sub}(G)$. In particular, our focus is on the case where $M(H)$ is a centralizer-like subgroup.

The number of subgroups, normal subgroups and characteristic subgroups of a finite group G are denoted by Sub(G), NSub(G) and CSub(G), respectively. The main goal of this paper is to present a matrix model for computing these positive integers for dicyclic groups, semi-dihedral groups, and three sequences U6n, V8n and H(n) of groups that can be presented as follows: U6n = ⟨a, b | a 2n = b 3 = e, bab = a⟩, V8n = ⟨a, b | a 2n = b 4 = e, aba = b −1 , ab−1a = b⟩, H(n) = ⟨a, b, c | a 2 n−2 = b 2 = c 2 = e, [a, b] = [b, c] = e, ac = ab⟩. For each group, a matrix model containing all information is given.