Sylow implies isomorph-conjugate

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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., Sylow subgroup) must also satisfy the second subgroup property (i.e., isomorph-conjugate subgroup)
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Statement

Any Sylow subgroup of a finite group is an isomorph-conjugate subgroup: it is conjugate to any isomorphic subgroup.

Related facts

Applications

Facts used

  1. Sylow implies order-conjugate: This is the conjugacy part of Sylow's theorem.
  2. Order-conjugate implies isomorph-conjugate

Proof

The proof follows directly from facts (1) and (2).