Direct factor not implies characteristic

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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

Analogues 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.