# Normal not implies finite-pi-potentially characteristic in finite

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a finite group, every subgroup satisfying the first subgroup property (i.e., normal subgroup) need not satisfy the second subgroup property (i.e., finite-pi-potentially characteristic subgroup)
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## Statement

It is possible to have a finite group $G$ and a normal subgroup $H$ of $G$ such that there is no finite group $K$ containing $G$ and satisfying both the following conditions:

• Every prime divisor of the order of $K$ is also a prime divisor of the order of $G$.
• $H$ is a characteristic subgroup of $K$.

## Facts used

1. Every finite p-group is a subgroup of a finite p-group that is not characteristic in any finite p-group properly containing it

## Proof

By fact (1), we can definitely find a finite $p$-group $H$ with the property that $H$ is not a characteristic subgroup in any finite $p$-group properly containing it.

Let $G$ be the direct product of $H$ and a cyclic group of order $p$. $H$ is thus a normal subgroup of $G$. However, for any finite $p$-group containing $G$, $H$ is a proper subgroup of it and therefore not a characteristic subgroup of it. Hence, $H$ cannot be made to be characteristic in any finite $p$-group $K$ that contains $G$.