Transitive normality is not quotient-transitive
This article gives the statement, and possibly proof, of a subgroup property (i.e., transitively normal subgroup) not satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property).
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Statement with symbols
It is possible to have the following situation: , is a transitively normal subgroup of a group , and is a transitively normal subgroup of , and is not a transitively normal subgroup of .
Transitively normal subgroup
Further information: Transitively normal subgroup
We say that is a transitively normal subgroup of if whenever is a normal subgroup of , is also a normal subgroup of .
The same example used here applies to all these related properties:
- Central factor is not quotient-transitive
- SCAB is not quotient-transitive
- Conjugacy-closed normality is not quotient-transitive
Example of the dihedral group
Further information: dihedral group:D8
Let be the dihedral group, given by:
is a subgroup of order two, hence it is transitively normal in . is a subgroup of order two in , hence it is transitively normal in .
However, is not transitively normal in , because the subgroup of is normal in but not in .