Normal Sylow satisfies transfer condition

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

Statement with symbols

Suppose $H$ is a normal Sylow subgroup of a finite group $G$ and $K\le G$ is any subgroup. Then $H\cap K$ is a normal Sylow subgroup of $K$.

Related facts

More on transfer condition for normal and Sylow subgroups

• Normality satisfies transfer condition
• Sylow does not satisfy transfer condition
• Hall does not satisfy transfer condition
• Normal Hall satisfies transfer condition

Intersecting a normal subgroup with a Sylow subgroup

• Equivalence of definitions of Sylow subgroup of normal subgroup: This states that a subgroup that is the intersection of a Sylow subgroup and a normal subgroup is a Sylow subgroup of the normal subgroup. Conversely, a subgroup expressible as a Sylow subgroup of a normal subgroup is expressible as an intersection of the normal subgroup with a Sylow subgroup of the whole group.

Facts used

1. Second isomorphism theorem
2. Lagrange's theorem

Proof

Given: A finite group $G$, a normal Sylow subgroup $H$, and a subgroup $K\le G$.

To prove: $H\cap K$ is a normal Sylow subgroup of $K$.

Proof: By the second isomorphism theorem, $H\cap K$ is normal in $K$, and we have:

$K/(H\cap K)\cong HK/H$.

Here, $HK$ is a subgroup of $G$. Since $G/H$ has order relatively prime to $p$, so does $HK/H$ (by fact (2)). Thus, $K/(H\cap K)$ has order relatively prime to $p$, so $[K:H\cap K]$ is relatively prime to $p$. Thus, $H\cap K$ is a normal $p$-subgroup of $K$ with index relatively prime to $p$, and is thus a normal $p$-Sylow subgroup of $K$.

References

Textbook references

• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 147, Exercise 32, Section 4.5 (Sylow's theorem)