Groups With Solvable Word Problems (Classic Reprint)

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Excerpt from Groups With Solvable Word ProblemsThis paper considers two aspects of the word problem for groups: first, some similarities between derivations from the relations of a group and proofs from a set of axioms in logic, and second, the computational difficulty of word problems and the problem of constructing groups with solvable word problems of some preassigned degree of difficulty.Given a presentation of a group in terms of generators and relations, we define two words on the generators to be equal if and only if one can be transformed into the other in a finite number of steps using the relations of the group. Defining equality in a group as derivability from a set of axioms suggests formulating the word problem for a group as the derivability problem for a formal system. The system we choose is equational logic. We show that there exist some striking analogies between results from logic and results about groups: nontrivial groups correspond to consistent systems, groups with solvable word problem to decidable systems, and simple groups to complete systems. This suggests that results about formal systems could be used to obtain results about decidability in groups.Hence there exist groups with arbitrarily difficult, but solvable word problem.About the PublisherForgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.comThis book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully, any imperfections that remain are intentionally left to preserve the state of such historical works.