Subgroup-defining function value is characteristic

Definition

Suppose $f$ is a subgroup-defining function, i.e., $f$ associates to every group $G$ a subgroup $f(G)$, with the property that given an isomorphism $\sigma:G_1 \to G_2$, the image of $f(G_1)$ under $\sigma$ is $f(G_2)$.

Then, for every group $G$, $f(G)$ is a characteristic subgroup of $G$.

Related facts

Conclusion about normality

Combined with the fact that characteristic implies normal, this tells us that any subgroup-defining function value is normal.

Applications to specific subgroup-defining functions

Subgroup-defining function	Meaning	Proof that it is characteristic	Proof of stronger properties satisfied for the particular subgroup-defining function
center	elements that commute with everything	center is characteristic	center is bound-word, center is strictly characteristic, center is direct power-closed characteristic
derived subgroup	products of commutators	derived subgroup is characteristic	derived subgroup is verbal, derived subgroup is fully invariant
Frattini subgroup	intersection of all maximal subgroups	Frattini subgroup is characteristic

Proof

Given: A group $G$, a subgroup-defining function $f$, an automorphism $\sigma$ of $G$.

To prove: $\sigma(f(G)) = f(G)$.

Proof: Since $\sigma:G \to G$ is an automorphism, it is in particular an isomorphism. Thus, the definition of subgroup-defining function above, setting $G_1 = G_2 = G$, gives us what we need to prove.