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)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about Sylow subgroup |Get facts that use property satisfaction of Sylow subgroup | Get facts that use property satisfaction of Sylow subgroup|Get more facts about intermediate subgroup condition

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 $p$.

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 $[K:H]$.