This publication presents the most essential portions of the author's German Diplomarbeit of 1984 and a "sprinkling" of new considerations and results as well which earnestly try to propose coming directions for the (timeless and eternal) Sylow theory in (locally) finite groups (see the pages 14, 15, 24 and xi to xxi). The line numbering provided facilitates and shall lively encourage (Scholarly Peer) Review. This edition proudly completes the Endnote g of the 3rd edition without line numbering (ISBN 978-3-7543-3213-9) with all the classical proofs and provides Appendix 1 and Appendix 2 for easy look at [1] and [8].