Sylow implies WNSCDIN

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 fact about::normal subsets of $$P$$. Then, if there exists $$g \in G$$ such that $$gAg^{-1} = B$$, there exists $$k \in N_G(P)$$ such that $$kAk^{-1} = B$$.

Similar facts

 * Sylow implies MWNSCDIN
 * Center of pronormal subgroup is subset-conjugacy-determined in normalizer
 * Center of Sylow subgroup is subset-conjugacy-determined in normalizer
 * Abelian Sylow implies SCDIN
 * Sylow and TI implies CDIN

Opposite facts

 * Sylow not implies CDIN

Facts used

 * 1) uses::Sylow implies pronormal
 * 2) uses::Pronormal implies WNSCDIN

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