Every subgroup is contracharacteristic in its normal closure

Statement
Any subgroup of a group is a contracharacteristic subgroup of its normal closure. In particular, it occurs as a contracharacteristic subgroup of a normal subgroup.

Contracharacteristic subgroup
A subgroup of a group is termed contracharacteristic if it is not contained in any proper characteristic subgroup.

Related facts

 * Characteristic of normal implies normal
 * Equivalence of definitions of subgroup of Abelian normal subgroup

Facts used

 * 1) uses::Characteristic of normal implies normal

Hands-on proof
Given: A subgroup $$H \le G$$, $$H^G$$ is the normal closure of $$H$$ in $$G$$.

To prove: If $$K \le H^G$$ is a characteristic subgroup of $$H^G$$ containing $$H$$, then $$K = H^G$$.

Proof:


 * 1) By fact (1), we see that since $$K$$ is characteristic in $$H^G$$ and $$H^G$$ is normal in $$G$$, we obtain that $$K$$ is normal in $$G$$.
 * 2) Thus, $$K$$ is a normal subgroup of $$G$$ containing $$H$$. By definition of normal closure, we get that $$H^G \le K$$. Since $$K \le H^G$$, we get $$K = H^G$$.