Characteristic not implies sub-(isomorph-normal 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., characteristic subgroup) need not satisfy the second subgroup property (i.e., sub-(isomorph-normal characteristic) subgroup)
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Statement

We can have a finite group $G$ and a characteristic subgroup $H$ of $G$ such that $H$ is not sub-(isomorph-normal characteristic) in $G$. In other words, there is no ascending chain of subgroups from $H$ to $G$ such that each member is an isomorph-normal characteristic subgroup of its successor.

Related facts

• Characteristic not implies sub-isomorph-free in finite
• Characteristic not implies isomorph-free in finite

Proof

The center of a non-abelian group of odd prime cube order

Further information: Prime-cube order group:U3p

Let $p$ be an odd prime. Let $G$ be the non-abelian group of order $p^3$ and exponent $p$. Let $H$ be the center of $G$. Then, we have:

• $H$ is characteristic in $G$.
• $H$ is not sub-(isomorph-normal characteristic) in $G$: In fact, no proper subgroup of $G$ containing $H$ is isomorph-normal and characteristic. $H$ itself is not isomorph-normal, because there are other cyclic groups of order $p$. Moreover, $H$ is a maximal characteristic subgroup of $G$, so there is no other possibility.