T-group property is normal subgroup-closed

Statement

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.

Definitions used

T-group

Further information: 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).