Normality is quotienttransitive
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Contents
Statement
Symbolic statement
Suppose is a normal subgroup of and is a subgroup of containing , such that is normal in . Then is also normal in .
Related facts
 Third isomorphism theorem is a statement describing the isomorphism types of the various quotient groups.
Generalization and other particular cases
A general version is:
Quotientbalanced implies quotienttransitive
Other particular cases are:
 Characteristicity is quotienttransitive
 Strict characteristicity is quotienttransitive
 Full invariance is quotienttransitive
Related facts about normality
 Normality is strongly joinclosed
 Normality is not transitive
 Normality satisfies image condition
 Normality satisfies inverse image condition
Proof
Handson proof
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 .