P-normal group

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The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
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Definition

Short version

Suppose G is a finite group and p is a prime number. We say that G is p-normal if the conjugacy functor on G arising from the characteristic p-functor sending a finite p-group to its center is a weakly closed conjugacy functor on G.

Long version

Suppose G is a finite group and p is a prime number. We say that G is p-normal if it satisfies the following equivalent conditions:

  1. Either of these equivalent:
  2. Either of these equivalent:
  3. Either of these equivalent:

Equivalence of definitions

Further information: equivalence of definitions of weakly closed conjugacy functor

The equivalence between both versions of (1), the equivalence between both versions of (2), and the equivalence between both versions of (3), follow from the fact that Sylow implies order-conjugate: any two p-Sylow subgroups are conjugate, and the conjugating automorphism preserves all properties including weak closure. The (1) implies (2) implies (3) direction is straightforward. The (3) implies (1) direction follows from the fact that characteristic subgroup of Sylow subgroup is weakly closed iff it is normal in every Sylow subgroup containing it.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
p-nilpotent group there is a p'-Hall subgroup, i.e., a normal p-complement. The p-Sylow subgroup is thus a retract. p-nilpotent implies p-normal p-normal not implies p-nilpotent |FULL LIST, MORE INFO
group in which the p-Sylow subgroup is abelian

Incomparable properties

References

  • The Theory of Groups by Marshall Hall, Jr., Page 205, Section 14.4, More info
  • Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 256, Chapter 7 (Fusion, transfer and p-factor groups), Section 6(Elementary applications),
    ",Chapter7(Fusion,transferandp-factorgroups),Section6(Elementaryapplications)" can not be assigned to a declared number type with value 256.
    ",Chapter7(Fusion,transferandp-factorgroups),Section6(Elementaryapplications)" can not be assigned to a declared number type with value 256.
    More info