Tame Sylow intersection

Definition

Suppose $G$ is a finite group and $p$ is a prime number dividing the order of $G$. Suppose, further, that $P,Q$ are (possibly equal) $p$-Sylow subgroups. The intersection $P\cap Q$ is termed a tame Sylow intersection if $N_{P}(P\cap Q)$ and $N_{Q}(P\cap Q)$ are both $p$-Sylow subgroups of $N_{G}(P\cap Q)$.

Note that this property depends on the specific choice of subgroups $P,Q$ we pick, and not just on their intersection. Specifically, we are interested in the embedding of $P\cap Q$ inside $P$ relative to $G$.

(Some definitions add a distinctness requirement from the two $p$-Sylow subgroups, in which case a $p$-Sylow subgroup ceases to be a tame intersection with itself).

Relation with other properties

Stronger properties

• Well-placed tame Sylow intersection: This places the added requirement that the intersection $P\cap Q$ be a well-placed subgroup inside $P$.
• Maximal Sylow intersection: For full proof, refer: Maximal Sylow intersection is tame

Weaker properties

• Subgroup whose normalizer in the Sylow is Sylow in the normalizer: By definition, the tame Sylow intersection has this property in both $P$ and $Q$.

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 240, Section 7.2 (Alperin's theorem)