Normality is quotient-transitive
This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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Suppose is a normal subgroup of and is a subgroup of containing , such that is normal in . Then is also normal in .
- Third isomorphism theorem is a statement describing the isomorphism types of the various quotient groups.
Generalization and other particular cases
A general version is:
Other particular cases are:
- Characteristicity is quotient-transitive
- Strict characteristicity is quotient-transitive
- Full invariance is quotient-transitive
Related facts about normality
- Normality is strongly join-closed
- Normality is not transitive
- Normality satisfies image condition
- Normality satisfies inverse image condition
Pick . Then the map is an inner automorphism of , hence sends to itself and induces an automorphism of . More importantly, the induced automorphism on is also an inner automorphism, by the image of under the quotient map . Since is normal in , the induced map on preserves the subgroup . Hence sends to .