A Sylow \(p\)-subgroup contains the full prime-power part available at \(p\) and is the standard local entry point for analyzing a finite group one prime at a time. [@isaacs2008]
Definition
For finite groups, being maximal among \(p\)-subgroups by inclusion is equivalent to having the full order \(p^a\). This is different from being an arbitrary maximal subgroup.
Core result
The full Wielandt existence proof, embedding and conjugacy arguments, and counting proof appear in the Sylow-theorems core article and Isaacs, Theorems 1.7 and 1.11-1.17.
Structural properties
- \(P\in\operatorname{Syl}_p(G)\) is normal exactly when \(n_p(G)=1\).
- \(n_p(G)=|G:N_G(P)|\), so Sylow counting is a normalizer-index calculation.
- If \(N\triangleleft G\), then \(P\cap N\in\operatorname{Syl}_p(N)\) and \(PN/N\) is Sylow in \(G/N\).
- \(O_p(G)\) is the intersection of all Sylow \(p\)-subgroups.
- A finite group is nilpotent exactly when every Sylow subgroup is normal, in which case it is their direct product.
Sylow subgroups need not be unique, but conjugacy makes them one global orbit. Normalizers, intersections, and fusion describe how these local models fit together.
Example and boundary
Knowledge network
The Sylow core article gives the complete proofs and applications, the p-core article treats their common normal part, and the normalizer article explains the count.
Proof and sources
The definition and four Sylow conclusions are Isaacs, Theorems 1.7 and 1.11-1.18; proofs are linked to the core article to avoid duplication. [@isaacs2008]
