Every subgroup is contracharacteristic in its normal closure

This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Contracharacteristic subgroup (?) and Normal subgroup (?)), to another known subgroup property (i.e., Subgroup (?))
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This fact is an application of the following pivotal fact/result/idea: characteristic of normal implies normal
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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.

Definitions used

Contracharacteristic subgroup

Further information: 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. Characteristic of normal implies normal

Proof

Hands-on proof

Given: A subgroup $H\leq G$, $H^{G}$ is the normal closure of $H$ in $G$.

To prove: If $K\leq H^{G}$ 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 $H^{G}\leq K$. Since $K\leq H^{G}$, we get $K=H^{G}$.