Central products and the Chermak-Delgado lattice (Q6597198)

Let \(G\) be a finite group. In [\textit{A. Chermak} and \textit{A. Delgado}, Proc. Am. Math. Soc. 107, No. 4, 907--914 (1989; Zbl 0687.20022)], a modular, self-dual sublattice \(\mathcal{CD}(G)\) of the lattice of subgroups of \(G\) is introduced, now universally known as the Chermak-Delgado lattice of \(G\). This lattice is defined using the measure \(m\) which takes subgroups of \(G\) to positive integers via the formula \(m(H)=|H| \cdot |C_{G}(H)|\) and considering the subgroups for which \(m(H)\) is maximal.\N\NThe main results in the paper under review are:\N\NTheorem A: Let \(G\) be a finite group and \(A, B \leq G\) such that \(G\) is a central product of \(A\) and \(B\). Then \(\mathcal{CD}(A)\cdot \mathcal{CD}(B) \subseteq \mathcal{CD}(G)\). Furthermore, the top (bottom) of \(\mathcal{CD}(G)\) is equal to the product of the tops (bottoms) of \(\mathcal{CD}(A)\) and \(\mathcal{CD}(B)\).\N\NTheorem B. If \(G\) is a central product of \(A\) and \(B\), then the height of the Chermak-Delgado lattice of \(G\) is equal to the sum of the heights of the Chermak-Delgado lattices of \(A\) and \(B\). Moreover, an element \(HK \in \mathcal{CD}(G)\) with \(A \cap B \leq H \leq A \) and \(A \cap B \leq K \leq B\) has height (depth) equal to the sum of the heights (depths) of \(H \in \mathcal{CD}(A)\) and \(K \in \mathcal{CD}(B)\).