P-automorphism-invariant subgroup

Definition
Let $$p$$ be a prime number. A subgroup $$H$$ of a p-group $$G$$ (i.e., a group where the order of every element is a power of $$p$$) is termed $$p$$-automorphism-invariant if, for any automorphism $$\sigma$$ of $$G$$ such that the order of $$\sigma$$ is a power of $$p$$, $$\sigma(H) = H$$.

For the property the context of finite $$p$$-groups, refer p-automorphism-invariant subgroup of finite p-group.

Equivalent properties
For a finite $$p$$-group, the property of being $$p$$-automorphism invariant equals the property of being a subnormal stability automorphism-invariant subgroup. That's because any $$p$$-automorphism can be realized as the stability automorphism of a subnormal series, and conversely, any stability automorphism of a subnormal series must be a $$p$$-automorphism.

Stronger properties

 * Characteristic subgroup:

Weaker properties

 * Stronger than::Normal subgroup