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

1. "On the influence of transitively normal subgroups on the structure of some infinite groups". Project Euclid. Retrieved 30 June 2022.

See also

• Normal subgroup