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, [math]\displaystyle{ H }[/math] is a transitively normal subgroup of [math]\displaystyle{ G }[/math] if for every [math]\displaystyle{ K }[/math] normal in [math]\displaystyle{ H }[/math], we have that [math]\displaystyle{ K }[/math] is normal in [math]\displaystyle{ G }[/math].^[1]

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.

References

See also

• Normal subgroup

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