Difference between revisions of "Nilpotent Hall implies isomorph-conjugate"

From Groupprops
Jump to: navigation, search
 
Line 10: Line 10:
 
==Facts used==
 
==Facts used==
  
[[Nilpotent Hall subgroups of same order are conjugate]]
+
# [[uses::Nilpotent Hall subgroups of same order are conjugate]]
 
+
# [[uses::Sylow implies intermediately isomorph-conjugate]]
 +
# [[uses::Hall implies join of Sylow subgroups]]
 +
# [[uses::Nilpotent join of intermediately isomorph-conjugate subgroups is intermediately isomorph-conjugate]]
 
==Related facts==
 
==Related facts==
  
Line 17: Line 19:
 
* [[Hall not implies isomorph-automorphic]]
 
* [[Hall not implies isomorph-automorphic]]
 
* [[Nilpotent Hall not implies order-conjugate]]
 
* [[Nilpotent Hall not implies order-conjugate]]
 +
 +
==Proof==
 +
 +
===Proof using fact (1)===
 +
 +
The proof follows directly from fact (1).
 +
 +
===Proof using facts (2)-(4)===
 +
 +
The proof follows directly by combining facts (2)-(4).

Latest revision as of 10:19, 29 September 2008

This article gives the statement and possibly, proof, of an 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., Nilpotent Hall subgroup (?)) must also satisfy the second subgroup property (i.e., Isomorph-conjugate subgroup (?)). In other words, every nilpotent Hall subgroup of finite group is a isomorph-conjugate subgroup of finite group.
View all subgroup property implications in finite groups | View all subgroup property non-implications in finite groups | View all subgroup property implications | View all subgroup property non-implications

Statement

Any Nilpotent Hall subgroup (?) (i.e., a Hall subgroup (?) that is also a Nilpotent group (?))of a finite group is isomorph-conjugate: it is conjugate to any isomorphic subgroup.

Facts used

  1. Nilpotent Hall subgroups of same order are conjugate
  2. Sylow implies intermediately isomorph-conjugate
  3. Hall implies join of Sylow subgroups
  4. Nilpotent join of intermediately isomorph-conjugate subgroups is intermediately isomorph-conjugate

Related facts

Proof

Proof using fact (1)

The proof follows directly from fact (1).

Proof using facts (2)-(4)

The proof follows directly by combining facts (2)-(4).