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 Mat. Zametki, 2010, Volume 88, Issue 6, Pages 803–810 (Mi mz8913)

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

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: 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.

Keywords: Maltsev problem, operations over groups, hereditory property for subgroups, Novikov–Adyan theory, simple groups

DOI: https://doi.org/10.4213/mzm8913

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English version:
Mathematical Notes, 2010, 88:6, 771–775

Bibliographic databases:

Document Type: Article
UDC: 512.54+512.54.0+512.543

Citation: S. I. Adian, “Once More on Periodic Products of Groups and on a Problem of A. I. Maltsev”, Mat. Zametki, 88:6 (2010), 803–810; Math. Notes, 88:6 (2010), 771–775

Citation in format AMSBIB
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This publication is cited in the following articles:
1. S. I. Adian, “The Burnside problem and related topics”, Russian Math. Surveys, 65:5 (2010), 805–855
2. V. S. Atabekyan, “On normal subgroups in the periodic products of S. I. Adian”, Proc. Steklov Inst. Math., 274 (2011), 9–24
3. Atabekyan V.S., Gevorgyan A.L., “On Outer Normal Automorphisms of Periodic Products of Groups”, J. Contemp. Math. Anal.-Armen. Aca., 46:6 (2011), 289–292
4. Atabekyan V.S., “On Cep-Subgroups of N-Periodic Products”, J. Contemp. Math. Anal.-Armen. Aca., 46:5 (2011), 237–242
5. A. L. Gevorgyan, “On automorphisms of periodic products of groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2012, no. 2, 3–9
6. S. S. Goncharov, Yu. L. Ershov, V. M. Levchuk, V. D. Mazurov, V. I. Senashov, A. I. Sozutov, N. S. Chernikov, “Vladimir Petrovich Shunkov (obituary)”, Russian Math. Surveys, 68:4 (2013), 769–771
7. S. I. Adian, V. S. Atabekyan, “The Hopfian Property of $n$-Periodic Products of Groups”, Math. Notes, 95:4 (2014), 443–449
8. S. I. Adian, Varuzhan Atabekyan, “Characteristic properties and uniform non-amenability of $n$-periodic products of groups”, Izv. Math., 79:6 (2015), 1097–1110
9. Adian S.I. Atabekyan V.S., “Periodic Products of Groups”, J. Contemp. Math. Anal.-Armen. Aca., 52:3 (2017), 111–117
10. Atabekyan V.S., Gevorgyan A.L., Stepanyan Sh.A., “The Unique Trace Property of N-Periodic Product of Groups”, J. Contemp. Math. Anal.-Armen. Aca., 52:4 (2017), 161–165
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