Classifying p-Groups via Homotopy Fibrations and Chebiam Decomposition

We introduce the Chebiam decomposition theorem, a novel structural result connecting the homotopy theory of classifying spaces BG for finite p-groups G with their group-theoretic properties. Our main result establishes that for any finite p-group G, the classifying space BG admits a canonical homotopy fibration decomposition that reflects the lower central series of G. This decomposition provides new computational tools for calculating cohomology rings H∗(BG; Fp) and leads to a classification of p-groups with equivalent homotopy types of classifying spaces. We prove that this decomposition is functorial and stable under certain group extensions, opening new avenues for understanding the algebraic topology of p-groups.