I-automorphism-invariance satisfies intermediate subalgebra condition

From Groupprops

Statement

Suppose $\mathcal{V}$ is a variety of algebras, $A$ is an algebra of $\mathcal{V}$ and $B$ is a subalgebra of $A$ such that for every I-automorphism $\varphi$ of $A$ , $\varphi(B) \subseteq B$ . Then, if $C$ is a subalgebra of $A$ containing $B$ , every I-automorphism of $C$ also sends $B$ to itself.

Related facts

Particular cases

Normality satisfies intermediate subgroup condition: For the variety of groups, the I-automorphisms are the inner automorphisms, and the subgroups invariant under these automorphisms are the normal subgroups. Thus, the statement translates to saying that a normal subgroup of a group is normal in every intermediate subgroup.