Maximal subgroup of Sylow subgroup

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Definition

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

• $H$ is a maximal subgroup of some $p$-Sylow subgroup of $G$.
• $H$ is a subgroup such that the largest power of $p$ dividing the index of $H$ in $G$ is $p$ itself.
• $H$ is a subgroup of order $p^{m-1}$ where $p^{m}$ is the largest power of $p$ dividing the order of $G$.

Relation with other properties

Stronger properties

• Maximal Sylow intersection: An intersection of two (usually distinct) Sylow subgroups that is a maximal subgroup in both.
• Characteristic maximal subgroup of Sylow subgroup

Weaker properties

• Normal subgroup of Sylow subgroup: $H$ is normal in any of the Sylow subgroups containing it.