# Contranormality is transitive

## Statement

### Verbal statement

A contranormal subgroup of a contranormal subgroup of a group is contranormal in the whole group.

## Facts used

1. Normality satisfies transfer condition: If $L$ is normal in $G$, and $K \le G$, then $K \cap L$ is normal in $K$.

## Proof

Given: A group $G$, a contranormal subgroup $K$ of $G$, a contranormal subgroup $H$ of $K$.

To prove: If $L$ is a normal subgroup of $G$ containing $H$, then $L = G$.

Proof: By fact (1), $L \cap K$ is normal in $K$, and contains $H$. Thus, since $H$ is contranormal in $K$, $L \cap K = K$. Thus, $K \le L$. So, $L$ is a normal subgroup of $G$ containing $K$. Since $K$ is contranormal in $G$, we get $L = G$.