We first give a quite profound overview of the structure of arbitrary simple groups and in particular of the simple locally finite groups and reduce their Sylow theory for the prime p to a famous conjecture of Prof. Otto H. Kegel ( see [16], Theorem 2.4: " Let P be a puniqueness subgroup of the finite simple group S which belongs to one of the 7 rankunbounded families. Then the rank of S will be bounded in terms of P ." ) about the rankunbounded ones of the 19 known families of finite simple groups. We introduce a new scheme to describe the 19 families, the family T of types, define the rank of each type, and emphasise the rôle of Kegel covers. This part presents a rather complete unified picture of known results all proofs of which are by reference and it is the actual reason why our title starts with "About ".We then apply new ideas to prove the conjecture for the alternating groups.Thereupon we are remembering Kegel covers and  -sequences. Finally we suggest a plan how to prove and even how to optimise the conjecture step-by-step or peu à peu which leads to further tough conjectures thereby unifying Sylow theory in locally finite simple groups with

This comprehensive review concerns the trilogy about Sylow Theory in Locally Finite Groups which has been published in four books by Books on Demand (BoD), Norderstedt, Germany, namely: Part 1 - ISBN 978-3-7543-6087-3 (November 2022); Part 1 - Second edition - ISBN 978-3-7568-0801-4 (March 2023); Part 2 - ISBN 978-3-7568-3892-9 (December 2022); Part 3 - ISBN 978-3-7568-9853-4 (January 2023).