Identity functor controls strong fusion for abelian Sylow subgroup

Statement in terms of subgroup properties
Suppose $$H$$ is an abelian Sylow subgroup of a finite group $$G$$. Then, $$H$$ is a SCDIN-subgroup of $$G$$, i.e., it is a subset-conjugacy-determined subgroup relative to its normalizer in $$G$$. Explicitly, given any two (possibly equal) subsets $$A,B$$ of $$H$$, and any element $$g$$ of $$G$$ such that $$gAg^{-1} = B$$, there exists $$k \in N_G(H)$$ such that $$kak^{-1} = gag^{-1}$$ for all $$a \in A$$.

Statement in terms of control of strong fusion
Suppose $$G$$ is a finite group and $$p$$ is a prime number. Suppose one (and hence every) $$p$$-Sylow subgroup of $$G$$ is an abelian Sylow subgroup. Consider the $$p$$-fusion in $$G$$. The identity functor, which is a function that sends every $$p$$-subgroup to itself, is a conjugacy functor that controls strong fusion in $$G$$. In other words, any bijection between subsets of a Sylow subgroup that is achieved via conjugation by an element of $$G$$ can also be achieved via conjugation by an element in the normalizer of the Sylow subgroup.

Fusion system version

 * Identity functor controls strong fusion for saturated fusion system on abelian group

Facts used

 * 1) uses::Sylow implies MWNSCDIN (this follows from uses::Sylow implies pronormal and uses::pronormal implies MWNSCDIN)
 * 2) uses::Abelian and MWNSCDIN implies SCDIN

Proof of statement in terms of subgroup properties
The proof follows directly by combining Facts (1) and (2).

Proof of statement in terms of control of strong fusion
This follows from the other version and the fact that both versions are saying the same thing.