Normality is quotient-transitive
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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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:
Quotient-balanced implies quotient-transitive
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
Proof
Hands-on 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
.