Sylow intersection

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Definition

Definition with symbols

Let $G$ be a finite group and $p$ be a prime number. A subgroup $H$ of $G$ is termed a $p$-Sylow intersection if it satisfies the following equivalent conditions:

• $H$ can be expressed as $P\cap Q$ for two distinct $p$-Sylow subgroups $P,Q$ of $G$.
• $H$ can be expressed as $P\cap N_{G}(Q)$ for two distinct $p$-Sylow subgroups $P,Q$ of $G$.

A subgroup is termed a Sylow intersection if it is a $p$-Sylow intersection for some $p$.

Equivalence of definitions

Further information: Equivalence of definitions of Sylow intersection

Relation with other properties

Stronger properties

• Tame Sylow intersection
• Maximal Sylow intersection

Weaker properties

• Intersection of Sylow subgroups