Weakly closed subgroup of Sylow subgroup

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Definition

As a property of a subgroup in a Sylow subgroup

Suppose $G$ is a finite group, $P$ is a $p$-Sylow subgroup of $G$ and $H$ is a subgroup of $P$. The triple $H\leq P\leq G$ denotes a weakly closed subgroup of Sylow subgroup if $H$ is a weakly closed subgroup of $P$ relative to $G$. In other words, any conjugate of $H$ by an element of $G$, that is contained in $P$, is equal to $H$.

As a subgroup property

A subgroup of a finite group is termed a weakly closed subgroup of Sylow subgroup if it satisfies the following equivalent conditions:

• It occurs as a weakly closed subgroup of a Sylow subgroup.
• It occurs as a weakly closed subgroup in every Sylow subgroup of the group containing it.

Relation with other properties

Stronger properties

• Sylow subgroup

Weaker properties

• p-subgroup that is normal in every p-Sylow subgroup containing it
• Normal subgroup of Sylow subgroup