Direct factor not implies characteristic

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., direct factor) need not satisfy the second subgroup property (i.e., characteristic subgroup)
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Statement

A direct factor of a group need not be a characteristic subgroup.

Related facts

Similar facts for subgroup properties

• Normal not implies characteristic
• Normal not implies direct factor
• Characteristic not implies direct factor

Analogues in Lie rings

• Direct factor not implies derivation-invariant
• Direct factor not implies characteristic in Lie rings

Proof

Example of a direct product

Let $G$ be any nontrivial group. Then consider $K=G\times G$, viz., the external direct product of $G$ with itself. The subgroups $G_{1}:=G\times \{e\}$ and $G_{2}:=\{e\}\times G$ are direct factors of $K$, and are hence both normal in $K$. Note also that they are distinct, since $G$ is nontrivial.

However, the exchange automorphism:

$(x,y)\mapsto (y,x)$

exchanges the subgroups $G_{1}$ and $G_{2}$. Thus, neither $G_{1}$ nor $G_{2}$ is invariant under all the automorphisms, so neither is characteristic. Thus, $G_{1}$ and $G_{2}$ are both direct factors of $K$ that are not characteristic.