Conjugacy-closedness is transitive

Statement with symbols
Suppose $$H \le K \le G$$ are groups such that $$H$$ is conjugacy-closed in $$K$$ and $$K$$ is conjugacy-closed in $$G$$.

Related facts

 * Conjugacy-closed normality is transitive
 * Central factor is transitive

Hands-on proof
Given: Groups $$H \le K \le G$$ such that $$H$$ is conjugacy-closed in $$K$$ and $$K$$ is conjugacy-closed in $$G$$.

To prove: $$H$$ is conjugacy-closed in $$G$$.

Proof: We need to show that if $$a,b \in H$$ are conjugate in $$G$$, then they are conjugate in $$H$$. First, observe that since $$ H \le K$$, $$a,b$$ are elements of $$K$$ conjugate in $$G$$. Since $$K$$ is conjugacy-closed in $$G$$, $$a,b$$ are conjugate in $$K$$.

Thus, $$a,b$$ are elements of $$H$$ that are conjugate in $$K$$. Since $$H$$ is conjugacy-closed in $$K$$, the elements $$a,b$$ are conjugate in $$H$$.