Finite p-group that is not characteristic in any finite p-group properly containing it

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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History

The existence of such groups, along with some examples and important facts, was established in a paper by Bettina Wilkens.

Definition

Suppose $p$ is a prime number and $P$ is a finite $p$-group, i.e., a group of prime power order where the underlying prime is $p$. We say that $P$ is a finite p-group that is not characteristic in any finite p-group properly containing it if, for any finite $p$-group $Q$ containing $P$, $P$ is not a characteristic subgroup (i.e., characteristic subgroup of group of prime power order) of $Q$.

If $P\leq Q$ and $P$ satisfies this property, and $P$ is proper in $Q$, then $P$ is not a p-finite-potentially characteristic subgroup of $Q$.

Facts

• Sylow subgroup of holomorph of cyclic group of prime-cube order is a finite p-group that is not characteristic in any finite p-group property containing it for odd prime
• Every finite p-group is a subgroup of a finite p-group that is not characteristic in any finite p-group properly containing it

References

Journal references

• Finite p-groups not characteristic in any finite p-group in which they are properly contained by Bettina Wilkens, , Volume 166, Page 97 - 112(Year 2008): ^{Official copy (PDF)(gated)More info}