Direct factor not implies characteristic

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

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

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.