Intermediately isomorph-conjugate of normal implies pronormal: Difference between revisions
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weaker = pronormal subgroup}} | weaker = pronormal subgroup}} | ||
{{composition computation|intermediately isomorph-conjugate subgroup|normal subgroup}} | {{composition computation| | ||
left = intermediately isomorph-conjugate subgroup| | |||
right = normal subgroup| | |||
final = pronormal subgroup}} | |||
==Statement== | ==Statement== |
Latest revision as of 19:28, 21 September 2008
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., intermediately isomorph-conjugate subgroup of normal subgroup) must also satisfy the second subgroup property (i.e., pronormal subgroup)
View all subgroup property implications | View all subgroup property non-implications
Get more facts about intermediately isomorph-conjugate subgroup of normal subgroup|Get more facts about pronormal subgroup
This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Intermediately isomorph-conjugate subgroup (?) and Normal subgroup (?)), to another known subgroup property (i.e., Pronormal subgroup (?))
View a complete list of composition computations
Statement
Property-theoretic statement
The subgroup property of being an intermediately isomorph-conjugate subgroup of normal subgroup (i.e., the subgroup property obtained by applying the composition operator to the properties intermediately isomorph-conjugate subgroup and normal subgroup) is stronger than the subgroup property of being a pronormal subgroup.
Verbal statement
Any intermediately isomorph-conjugate subgroup of a normal subgroup of a group is pronormal.
Related facts
Weaker facts
Corollaries
- Sylow of normal implies pronormal
- Frattini's argument is a slight weakening of this.
Similar facts
- Intermediately automorph-conjugate of normal implies weakly pronormal
- Intermediately normal-to-characteristic of normal implies intermediately subnormal-to-normal
Converse
Most natural choices of converse to this statement aren't true. Specifically, it is not necessary that if is such that whenever is normal in , is pronormal in , then is intermediately isomorph-conjugate in . What we can guarantee for is that it is a procharacteristic subgroup of . Further information: Left residual of pronormal by normal is procharacteristic
Definitions used
Intermediately isomorph-conjugate subgroup
Further information: Intermediately isomorph-conjugate subgroup
Pronormal subgroup
Further information: Pronormal subgroup
Procharacteristic subgroup
Further information: Procharacteristic subgroup
Facts used
- Intermediately isomorph-conjugate implies procharacteristic
- Procharacteristic of normal implies pronormal
Proof
The proof follows by combining facts (1) and (2).