Sylow implies WNSCDIN

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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., WNSCDIN-subgroup)
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Statement

Verbal statement

Any Sylow subgroup of a finite group is a WNSCDIN-subgroup.

Statement with symbols

Suppose P is a p-Sylow subgroup of a finite group G, and A,B \subseteq P are Normal subset (?)s of P. Then, if there exists g \in G such that gAg^{-1} = B, there exists k \in N_G(P) such that kAk^{-1} = B.

Related facts

Similar facts

Opposite facts

Facts used

  1. Sylow implies pronormal
  2. Pronormal implies WNSCDIN

Proof

Proof using given facts

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

Hands-on proof

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