Splitting the center of a Sylow subgroup

TL;DR

The paper investigates the structure of Sylow p-subgroups in finite groups, proving a decomposition of their centers under normality conditions and extending the results to fusion systems, with some exceptions for p=2.

Contribution

It generalizes a known result about Sylow subgroup centers to broader classes of groups and fusion systems, identifying conditions for the decomposition.

Findings

Z(S) decomposes as a direct product under normality

Counterexamples exist for p=2 in non-solvable groups

Extension of results to fusion systems

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.