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

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Statement

Property-theoretic statement

The left transiter of the property of being a Normal subgroup of p-group (?) is the property of being a 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.