Normality satisfies transfer condition

Verbal statement
If a subgroup is normal in the group, its intersection with any other subgroup is normal in that subgroup.

Symbolic statement
Let $$H \le G$$ be a normal subgroup and let $$K$$ be any subgroup of $$G$$. Then, $$H \cap K \triangleleft K$$.

Property-theoretic statement
The subgroup property of being normal satisfies the transfer condition.

Normal subgroup
A subgroup $$H$$ of a group $$G$$ is said to be normal if for any $$g \in G$$ and $$h \in H$$, $$ghg^{-1} \in H$$.

Transfer condition
A subgroup property $$p$$ is said to satisfy transfer condition if whenever $$H, K$$ are subgroups of $$G$$ and $$H$$ has property $$p$$ in $$G$$, $$H \cap K$$ has property $$p$$ in $$K$$.

Further facts

 * Second isomorphism theorem: This result equates the quotient of the non-normal subgroup, by the intersection, with the quotient of the product of subgroups, by the normal subgroup. Specifically, it states that if $$H$$ is normal in $$G$$ and $$K$$ is any subgroup of $$G$$, we have $$H/(H \cap K) \cong HK/K$$.

Related metaproperties satisfied by normality

 * Stronger than::Normality satisfies intermediate subgroup condition: The intermediate subgroup condition is weaker. It says that if $$H \le K \le G$$ are subgroups are $$H$$ is normal in $$G$$, then $$H$$ is normal in $$K$$.
 * Weaker than::Normality satisfies inverse image condition: The inverse image condition is stronger. It says that the inverse image of a normal subgroup under a homomorphism is normal.

Other metaproperties satisfied by normality, that are somewhat related:


 * Normality satisfies image condition: The image of a normal subgroup under a surjective homomorphism is normal in the image.
 * Normality is upper join-closed

Transfer condition for other subgroup properties

 * Subnormality satisfies transfer condition: This follows directly from the fact that normality satisfies the transfer condition, and the fact that transfer condition is subordination-closed.
 * Permutability satisfies transfer condition

Analogues in other algebraic structure

 * Ideal property satisfies transfer condition in Lie rings

Hands-on proof
Given: A group $$G$$, a normal subgroup $$H \triangleleft G$$ and a subgroup $$K \le G$$

To prove: $$H \cap K \triangleleft K$$. In other words, we need to prove that given any $$g \in K$$ and $$h \in H \cap K$$, $$ghg^{-1} \in H \cap K$$.

Proof: Since $$h \in H \cap K$$, we in particular have $$h \in H$$. Since $$H \triangleleft G$$ (viz $$H$$ is normal in $$G$$), $$ghg^{-1} \in H$$.

But we also have that $$g \in K$$ and $$h \in K$$. Since $$K$$ is a subgroup, $$ghg^{-1} \in K$$.

Combining these two facts, $$ghg^{-1} \in H \cap K$$.

Textbook references

 * , Page 88, Exercise 24
 * , Page 53, Problem 5