The Strong Sylow Theorem for the Prime p in Projective Special Linear Locally Finite Groups - Part 3 of a Trilogy

In Part 3 of the Trilogy „Characterising Locally Finite Groups Satisfying the Strong Sylow Theorem for the Prime p“ & „About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups“ & „The Strong Sylow Theorem for the Prime p in Projective Special Linear Locally Finite Groups“ we continue the program begun in [10] to optimise along the way 1) its beautiful Theorem about the first type „An“ of infinite families of finite simple groups step-by-step to further types by proving it for the second type „A = PSL n“. We start with proving the Conjecture 2 of [10] about the General Linear Groups over (commutative) locally finite fields, stating that their rank is bounded in terms of their p-uniqueness, and then break down this insight to the Special Linear Groups and the Projective Special Linear (PSL) Groups over locally finite fields. We close with suggestions for future research regarding the remaining five rank-unbounded types (the „Classical Groups“) and the way 2), regarding the (locally) finite and p-soluble groups, and regarding Augustin-Louis Cauchy's and Évariste Galois' contributions to Sylow theory in finite groups.