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

This article is cited in 10 scientific papers (total in 11 papers)

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

S. I. Adian

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


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

Bibliographic databases:

UDC: 512.54+512.54.0+512.543
Received: 11.05.2010

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
\by S.~I.~Adian
\paper Once More on Periodic Products of Groups and on a Problem of A.~I.~Maltsev
\jour Mat. Zametki
\yr 2010
\vol 88
\issue 6
\pages 803--810
\jour Math. Notes
\yr 2010
\vol 88
\issue 6
\pages 771--775

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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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. V. S. Atabekyan, “On normal subgroups in the periodic products of S. I. Adian”, Proc. Steklov Inst. Math., 274 (2011), 9–24  mathnet  crossref  mathscinet  isi  elib  elib
    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  crossref  zmath  isi  scopus
    4. Atabekyan V.S., “On Cep-Subgroups of N-Periodic Products”, J. Contemp. Math. Anal.-Armen. Aca., 46:5 (2011), 237–242  crossref  mathscinet  zmath  isi  scopus
    5. A. L. Gevorgyan, “On automorphisms of periodic products of groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2012, no. 2, 3–9  mathnet
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    7. S. I. Adian, V. S. Atabekyan, “The Hopfian Property of $n$-Periodic Products of Groups”, Math. Notes, 95:4 (2014), 443–449  mathnet  crossref  crossref  mathscinet  isi  elib
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    9. Adian S.I. Atabekyan V.S., “Periodic Products of Groups”, J. Contemp. Math. Anal.-Armen. Aca., 52:3 (2017), 111–117  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    11. V. S. Atabekyan, L. D. Beklemishev, V. S. Guba, I. G. Lysenok, A. A. Razborov, A. L. Semenov, “Questions in algebra and mathematical logic. Scientific heritage of S. I. Adian”, Russian Math. Surveys, 76:1 (2021), 1–27  mathnet  crossref  crossref  mathscinet  isi  elib
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