Chermak–Delgado theory measures both size and centrality through \(m_G(H)=|H||C_G(H)|\). The subgroups of maximum measure form a self-dual modular lattice and yield a characteristic abelian subgroup with a sharp index-square bound. [@isaacs2008]
Definition
On \(CD(G)\) the map \(H\mapsto C_G(H)\) reverses inclusion and satisfies the double-centralizer identity \(C_G(C_G(H))=H\).
Core result
The key inequality is \(m_G(H)m_G(K)\leq m_G(H\cap K)m_G(\langle H,K\rangle)\). At maximum measure it forces intersections and joins to remain in the lattice and yields modularity. Centralizer duality gives the unique minimum and maximum. See Isaacs, Theorems 1.42-1.45.
Structural properties
- If \(H\in CD(G)\), then \(C_G(H)\in CD(G)\) and double centralization recovers \(H\).
- The lattice is invariant under every automorphism of \(G\), so its unique minimum is characteristic.
- The minimum \(M\) and its centralizer both lie in the lattice; minimality forces \(M\leq C_G(M)\), hence \(M\) is abelian.
- If some \(m_G(H)>|G|\), then \(G\) cannot be nonabelian simple because the maximum-measure lattice yields nontrivial characteristic structure.
- The construction generalizes Brodkey's index-square estimate for abelian Sylow subgroups without a Sylow hypothesis.
The measure turns the tension between a large subgroup and a large centralizer into lattice structure. It does not classify the group directly but reliably creates a canonical characteristic subgroup.
Example and boundary
Knowledge network
Centralizers provide the duality, characteristic subgroups explain canonicity, and Fitting theory supplies a different canonical construction based on normal nilpotence.
Proof and sources
The definition, product inequality, lattice structure, and index-square theorem are Isaacs, Theorems 1.41-1.46; this article records their exact conclusions and proof route. [@isaacs2008]
