The \(p\)-core \(O_p(G)\) gathers all normal \(p\)-structure in a finite group. It is the prime component of the Fitting subgroup and a basic tool for detecting nonsimplicity and the faithfulness of local actions. [@isaacs2008]
Definition
For a set of primes \(\pi\), \(O_\pi(G)\) is the largest normal \(\pi\)-subgroup. The notation \(O_{p'}(G)\) denotes the largest normal subgroup whose order is not divisible by \(p\).
Core result
A normal \(p\)-subgroup lies in every Sylow \(p\)-subgroup. Conversely, the intersection of all Sylow subgroups is conjugation-invariant and a \(p\)-group. The subnormal statement follows by taking normal closures one level at a time; see Isaacs, Problem 2A.1.
Structural properties
- \(O_p(G)\operatorname{char}G\) because automorphisms permute the normal \(p\)-subgroups.
- If \(N\triangleleft G\), then \(O_p(N)\leq O_p(G)\), while in quotients one generally has only \(O_p(G)N/N\leq O_p(G/N)\).
- \(F(G)=\prod_pO_p(G)\), and normal cores for distinct primes centralize one another.
- Every \(O_p(G)\) is trivial in a nonabelian simple group; the converse is false for many semisimple and almost simple groups.
- Kernels of conjugation actions are often candidates for \(O_p(G)\); excluding a normal \(p\)-core proves faithfulness.
\(O_p(G)\) differs from a Sylow subgroup: Sylow subgroups are maximal by order, while the \(p\)-core is maximal under global normality. They coincide exactly when a Sylow \(p\)-subgroup is normal.
Example and boundary
Knowledge network
The Fitting subgroup multiplies all \(p\)-cores; subnormality explains why subnormal local subgroups lie in a core; Sylow theory supplies the intersection characterization.
Proof and sources
The Sylow-intersection and Brodkey discussion is in Section 1F, and containment of subnormal \(p\)-subgroups is Problem 2A.1. Full proofs are linked from the Fitting and subnormal core articles. [@isaacs2008]
