1. Introduction 3 I. Background Materials 5 2. Words, free groups, and morphisms 6 3. Diagrams of A Group 8 1. Cell Decomposition 8 2. Diagram 11 3. Van Kampen’s Lemma 14 4. 0-Refinement 15 5. Orientable Diagram 17 6. Example: apply van Kampen’s lemma to the group 17 4. Graded Presentation 20 1. Cells of Rank i 20 2. Equivalent in Rank i, Conjugate in Rank i 21 3. Why Graded? 21 5. Contiguity Maps 23 1. Connecting Line 23 2. Adjacent Edges 23 3. 0-Bonds and 0-Contiguity Submaps 25 4. k-Bonds and k-Contiguity Submaps 27 5. Cancellable Cells and Reduced Diagrams 29 6. Diagrams We Need 31 7. l-aperiodic 34 8. The Parameters and Lowest Parameter Principle 35 II. The Construction of a Group with Subgroups of Bounded Order 36 9. A-map 37 1. Condition A 37 2. A Graded Presentation Under Condition A 38 10. Condition R 42 1. Partition of Relators for Condition R 42 2. Conditions R 44 III. Some Properties of the Group We Need 46 IV. Proof of the Theorem 1 49