Finite p-group that is not characteristic in any finite p-group properly containing it
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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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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
History
The existence of such groups, along with some examples and important facts, was established in a paper by Bettina Wilkens.
Definition
Suppose is a prime number and is a finite -group, i.e., a group of prime power order where the underlying prime is . We say that is a finite p-group that is not characteristic in any finite p-group properly containing it if, for any finite -group containing , is not a characteristic subgroup (i.e., characteristic subgroup of group of prime power order) of .
If and satisfies this property, and is proper in , then is not a p-finite-potentially characteristic subgroup of .
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}