Nilpotent Hall implies isomorph-conjugate

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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.
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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 using fact (1)

The proof follows directly from fact (1).

Proof using facts (2)-(4)

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