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

From Groupprops

## 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 is a p-group, i.e., a group in which the order of every element is a power of a fixed prime number , and is a subgroup of . Then, the following are equivalent:

- For any p-group containing as a normal subgroup, is also a normal subgroup of .
- is a p-automorphism-invariant subgroup of .

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