Sylow satisfies intermediate subgroup condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., Sylow subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)
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Statement

A Sylow subgroup of a finite group is also a Sylow subgroup in any intermediate subgroup.

Definitions used

Sylow subgroup

Further information: Sylow subgroup

A subgroup $P$ of a group $G$ is a Sylow subgroup if $P$ is a group of prime power order and the order of $P$ is relatively prime to the index of $P$.

$P$ is termed a $p$-Sylow subgroup if its order is a power of the prime $p$ and its index is not a multiple of $p$.

Related facts

• Hall satisfies intermediate subgroup condition
• Sylow does not satisfy transfer condition

Facts used

1. Index is multiplicative

Proof

Given: A finite group $G$, a $p$-Sylow subgroup $H$ of $G$, a subgroup $K$ of $G$ containing $H$.

To prove: $H$ is a Sylow subgroup (in fact, a $p$-Sylow subgroup) inside $K$.

Proof: Clearly, $H$ continues to be a $p$-subgroup inside $K$, because it is a $p$-group. Thus, it suffices to show that the index $[K:H]$ is relatively prime to $p$. For this, observe that by fact (1):

$[G:H]=[G:K][K:H]$

Thus, $[K:H]$ is a divisor of $[G:H]$. Since $[G:H]$ is relatively prime to $p$, so is $[K:H]$.