Normal Sylow satisfies transfer condition

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 $$K$$.

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) uses::Second isomorphism theorem
 * 2) uses::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 $$K$$.

Textbook references

 * , Page 147, Exercise 32, Section 4.5 (Sylow's theorem)