Normality satisfies transfer condition
This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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If a subgroup is normal in the group, its intersection with any other subgroup is normal in that subgroup.
Let be a normal subgroup and let be any subgroup of . Then, .
A subgroup property is said to satisfy transfer condition if whenever are subgroups of and has property in , has property in .
- 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 is normal in and is any subgroup of , we have .
Related metaproperties satisfied by normality
- Normality satisfies intermediate subgroup condition: The intermediate subgroup condition is weaker. It says that if are subgroups are is normal in , then is normal in .
- 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
Given: A group , a normal subgroup and a subgroup
To prove: . In other words, we need to prove that given any and , .
Proof: Since , we in particular have . Since (viz is normal in ), .
But we also have that and . Since is a subgroup, .
Combining these two facts, .