1-automorphism-invariant subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a $$1$$-automorphism-invariant subgroup if any 1-automorphism of $$G$$ sends $$H$$ to itself. In other words, for every 1-automorphism $$\varphi$$ of $$G$$, $$\varphi(H) \subseteq H$$.

Formalisms
The property of being a 1-automorphism-invariant subgroup can be expressed as an :

1-automorphism $$\to$$ Function

In other words, $$H$$ is a 1-automorphism-invariant subgroup of a group $$G$$ if and only if every 1-automorphism of $$G$$ restricts to a function from $$H$$ to itself.

Alternative function restriction expressions are:


 * 1-automorphism $$\to$$ 1-endomorphism
 * 1-automorphism $$\to$$ 1-automorphism. This shows that the property of being a 1-automorphism-invariant subgroup is a with respect to 1-automorphisms.

Examples

 * In any group, the whole group and the trivial subgroup are 1-automorphism-invariant.
 * In a cyclic group, every subgroup is 1-automorphism-invariant.
 * If the set of elements of order $$n$$ form a subgroup for any particular $$n$$, then that subgroup is 1-automorphism-invariant. This is because 1-automorphisms preserve the orders of elements.

Stronger properties

 * Weaker than::1-endomorphism-invariant subgroup

Weaker properties

 * Stronger than::Quasiautomorphism-invariant subgroup
 * Stronger than::Strong quasiautomorphism-invariant subgroup
 * Stronger than::Characteristic subgroup
 * Stronger than::Normal subgroup