Splitting the center of a Sylow subgroup

TL;DR

Let $p$ be a prime and $S$ a Sylow $p$-subgroup of a finite group $G$. The paper studies the splitting of $Z(S)$ over a subgroup closely related to $Z(S)\cap Z(G)$ by introducing the subgroup $W_G(S)$ of weakly closed elements and proving that, under broad hypotheses (notably when $p$ is odd, or $Z(S)\le O_{2',2}(G)$, or $G$ is solvable), $Z(S)$ splits as $\ker(\operatorname{tr}_S^H)\times W_G(S)$ with $W_G(S)$ a direct factor, and $W_G(S)=W_H(S)$ for suitable $H$. It then derives the analogous fusion-system result: for odd $p$ or $Z(S)\le O_p(\mathcal{F})$, the center of the fusion system $Z(\mathcal{F})$ is a direct factor of $Z(S)$, matching the group-theoretic case. Counterexamples at $p=2$ with non-solvable groups are provided, and the framework extends to saturated fusion systems, linking the splitting phenomenon to coprime action and transfer arguments.

Abstract

Suppose $p$ is a prime and $S$ is a Sylow $p$-subgroup of a finite group $G$. If $S$ is normal in $G$, then $Z(S)$ is the direct product of $S \cap Z(G)$ with $[Z(S), G]$. We prove an analogous result for all groups except in some cases where $p=2$ and $G$ is not solvable, where we have counterexamples. We also extend this result to fusion systems.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3
• Theorem 2.4: Glauberman1971
• proof : Proof of Theorem \ref{['thm WGS']}
• Example 2.5