T-group property is normal subgroup-closed

Property-theoretic statement
The group property of being a T-group satisfies the group metaproperty of being normal subgroup-closed.

Verbal statement
Any normal subgroup of a T-group is a T-group.

T-group
A T-group is a group in which normality is transitive: every normal subgroup of a normal subgroup (i.e. every 2-subnormal subgroup) is normal. Equivalently, every subnormal subgroup is normal.

Generalizations

 * Transitive implies intermediate subgroup condition implies closed under former

Proof
Given: A T-group $$G$$, a normal subgroup (or equivalently, a subnormal subgroup) $$N$$ of $$G$$

To prove: $$N$$ is a T-group

Proof: We need to show that if $$H$$ is a subnormal subgroup of $$N$$, then $$H$$ is normal in $$N$$. The proof uses the following steps:


 * Since $$H$$ is subnormal in $$N$$ and $$N$$ is normal in $$G$$, $$H$$ is subnormal in $$G$$
 * Since $$G$$ is a T-group, and $$H$$ is subnormal in $$G$$, $$H$$ is normal in $$G$$
 * Since normality satisfies intermediate subgroup condition, and $$H$$ is normal in $$G$$, $$H$$ is normal in $$N$$

This completes the proof. The proof follows a general pattern (see generalizations above).