Certain Simple, Unsolvable Problems of Group Theory. V 29,3029Parts I, II, III and IV have appeared in Series A, 57, Nos. 3 and 5; 58, Nos. 2 and 5 of these Proceedings, as well as Indag. Math., 16, Nos. 3 and 5 (1954); 17, Nos. 2 and 5 (1955).In Part I, page 234, the ninth and tenth lines following the displayed material, for both occurrences of A read D and for both occurrences of B read E: In part II, page 497, for all occurrences of M in Diagrams A and D read C. In Part IV, page 574, the third displayed line, for zαiLread z―αiL. 19).30The proof of the unsolvability of the word problem contained in this and subsequent paper is not contained in the dissertation Several simple unsolvable problems of group theory related to the word problem. (See footnote 1 of Part I.) Our proof was finally completed during the period 1954-56 while in residence at the Institute for Advanced Study. We were supported in 1954-55 directly by the Institute and in 1955-56 by National Science Foundation contract G-1974. Certain improvements were evolved ans preparations for publication were completed while the author held a Fulbright grant to the University of Oslo.Our thanks are due to Professor Kurt Gödel for his kind encouragement in these matter; certain related problems for study he has suggested we hope to deal with later. When an earlier version of Lemmas 30 and 31 was explained at a colloquium at the University of Michigan, August 3, 1956, Professor Roger Lyndon suggested an improvement which is incorporated in our present version. We are indeed indebted to Dr. John Addison and Dr. Michael Rabin for checking many of the new details during July of 1956.)