In the paper, the solvability of the word problem and the conjugacy problem is proved for a wide class of finitely presented groups defined by periodic defining relations of a sufficiently large odd degree. In the proof, we use a certain simplified version of the classification of periodic words and transformations of these words, which was presented in detail in the author's monograph devoted to the well-known Burnside problem. The result is completed by the proof of an interesting result of Sarkisyan on the existence of a group, given by defining relations of the form E-i(2) = 1, for which the word problem is unsolvable. This result was first published in abstracts of papers of the 13th All-Union Algebra Symposium in Gomel in 1975.
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