Transitively normal subgroup
- Transitively normal subgroup
In mathematics, in the field of group theory, a subgroup of a group is said to be transitively normal in the group if every normal subgroup of the subgroup is also normal in the whole group. In symbols, is a transitively normal subgroup of if for every normal in , we have that is normal in .
An alternate way to characterize these subgroups is: every "normal subgroup preserving automorphism" of the whole group must restrict to a "normal subgroup preserving automorphism" of the subgroup.
Here are some facts about transitively normal subgroups:
*Every normal subgroup of a transitively normal subgroup is normal.
*Every direct factor, or more generally, every central factor is transitively normal. Thus, every
central subgroup is transitively normal.
*A transitively normal subgroup of a transitively normal subgroup is transitively normal.
*A transitively normal subgroup is normal.
Also see: Normal subgroup
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