Once More on Periodic Products of Groups and on a Problem of A. I. Maltsev

A new operation of product of groups, the $n$-periodic product of groups for odd exponent $n\ge 665$, was proposed by the author in 1976 in the paper [1]. This operation is described on the basis of the Novikov–Adyan theory introduced in the monograph [2] of the author. It differs from the classic operations of direct and free products of groups, but has all of the natural properties of these operations, including the so-called hereditary property for subgroups. Thus, the well-known problem of A. I. Maltsev on the existence of such new operations was solved. Unfortunately, in the paper [1], the case where the initial groups contain involutions, was not analyzed in detail. It is shown that, in the case where the initial groups contain involutions, this small gap is easily removed by an additional restriction on the choice of defining relations for the periodic product. It suffices to simply exclude products of two involutions of previous ranks from the inductive process of defining new relations for any given rank $\alpha$. It is suggested that the adequacy of the given restriction follows easily from the proof of the key Lemma II.5.21 in the monograph [2]. We also mention that, with this additional restriction, all the properties of the periodic product given in [1] remain true with obvious corrections of their formulation. Moreover, under this restriction, one can consider $n$-periodic products for any period $n\ge665$, including even periods.