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This is an English translation of the thesis written by G. S. Makanin for thedegree of Candidate of Physical and Mathematical Sciences (equivalent to aPh.D.), originally submitted to the Steklov Mathematical Institute in 1966. Theoriginal language is Russian. The named supervisors are A. A. Markov and S. I.Adian.

A translation of G. S. Makanin's 1966 Ph.D. thesis "On the Identity Problem for Finitely Presented Groups and Semigroups"