Left transiter of normal is p-automorphism-invariant in p-groups

Property-theoretic statement
The left transiter of the property of being a fact about::normal subgroup of p-group is the property of being a fact about::p-automorphism-invariant subgroup.

The finite analogue also holds: the left transiter of the property of being a normal subgroup of group of prime power order is the property of being a p-automorphism-invariant subgroup of finite p-group.

Statement with symbols
Suppose $$G$$ is a p-group, i.e., a group in which the order of every element is a power of a fixed prime number $$p$$, and $$H$$ is a subgroup of $$G$$. Then, the following are equivalent:


 * 1) For any p-group $$K$$ containing $$G$$ as a normal subgroup, $$H$$ is also a normal subgroup of $$K$$.
 * 2) $$H$$ is a p-automorphism-invariant subgroup of $$K$$.

Also, we can replace $$p$$-group above by finite $$p$$-group everywhere.