Metasymmetric and metaalternating groups of infinite rank  (in Ukrainian)

B.V.Olijnyk; V.S.Sikora; V.I.Sushchanskiy

We characterize the structure of infinitely iterated wreath products of finite symmetric and alternating groups. The orders of these groups as profinite groups are calculated. A method of constructing finite generating systems of metaalternating groups is described. We construct explicit examples of such generating systems. The normalizer of a metaalternating group in the corresponding metasymmetric group is characterized. We establish that under a natural condition the metaalternating group is perfect.

1. Заводя М.В., Сікора В.С., Сущанський В.І. Системи твірних метазнакозмінних груп скінченного рангу// Наук. вiсн. Чернiвецького ун-ту: Зб. наук. праць. Математика.--- Чернiвцi: Рута, 2006.--- Вип. 314-315.--- С.64--72.

2. Cикора В.С., Сущанский В.И. Системы порождающих групп автоматных подстановок// Кибернетика и системный анализ.--- 2000.--- №3.--- С.121--133.

3. Bhattacharjee M. The probability of generating certain profinite groups by two elements// Israel Journal of Mathematics.--- 1994.--- V.86.--- P.311--329.

4. Quick M. Probabilistic generation of wreath products of non-abelian simple groups// Comm. Algebra.--- 2004.--- V.32.--- P.4753--4768.

5. Quick M. Probabilistic generation of wreath products of non-abelian simple groups. II// Internat. J. of Algebra and Comput.--- 2006.--- V.16.--- №3.--- P.493--503.

6. Woryna A. On generation of wreath products of cyclic groups by two-state time-varying Mealy automata// Internat. J. of Algebra and Comput.--- 2006.--- V.16.--- №2.--- P.397--415.

7. Wilson J. Profinite groups.--- Oxford: Clarendon Press, 1998.--- 284~p.

8. Conder M. More on generators for alternating and symmetric groups// Quarterly. J. of Mathematics.--- 1981.--- V.32.--- P.137--163.

9. Lubotsky A., Wilson J.S. An embedding theorem for profinite groups// Arch. Math. (Basel).--- 1984.--- V.53.--- P.397--399.

	Metasymmetric and metaalternating groups of infinite rank (in Ukrainian)
Author	B.V.Olijnyk, V.S.Sikora, V.I.Sushchanskiy Kyiv Mohyla Academy, Kyiv, Yuriy Fedkovych Chernivtsi National University, Chernivtsi, Silesian University of Technology, Glivice, Poland
Abstract	We characterize the structure of infinitely iterated wreath products of finite symmetric and alternating groups. The orders of these groups as profinite groups are calculated. A method of constructing finite generating systems of metaalternating groups is described. We construct explicit examples of such generating systems. The normalizer of a metaalternating group in the corresponding metasymmetric group is characterized. We establish that under a natural condition the metaalternating group is perfect.
Keywords	metasymmetric group, metalternating group, infinite rank
DOI	doi:10.30970/ms.29.2.139-150
Reference	1. Заводя М.В., Сікора В.С., Сущанський В.І. Системи твірних метазнакозмінних груп скінченного рангу// Наук. вiсн. Чернiвецького ун-ту: Зб. наук. праць. Математика.--- Чернiвцi: Рута, 2006.--- Вип. 314-315.--- С.64--72. 2. Cикора В.С., Сущанский В.И. Системы порождающих групп автоматных подстановок// Кибернетика и системный анализ.--- 2000.--- №3.--- С.121--133. 3. Bhattacharjee M. The probability of generating certain profinite groups by two elements// Israel Journal of Mathematics.--- 1994.--- V.86.--- P.311--329. 4. Quick M. Probabilistic generation of wreath products of non-abelian simple groups// Comm. Algebra.--- 2004.--- V.32.--- P.4753--4768. 5. Quick M. Probabilistic generation of wreath products of non-abelian simple groups. II// Internat. J. of Algebra and Comput.--- 2006.--- V.16.--- №3.--- P.493--503. 6. Woryna A. On generation of wreath products of cyclic groups by two-state time-varying Mealy automata// Internat. J. of Algebra and Comput.--- 2006.--- V.16.--- №2.--- P.397--415. 7. Wilson J. Profinite groups.--- Oxford: Clarendon Press, 1998.--- 284~p. 8. Conder M. More on generators for alternating and symmetric groups// Quarterly. J. of Mathematics.--- 1981.--- V.32.--- P.137--163. 9. Lubotsky A., Wilson J.S. An embedding theorem for profinite groups// Arch. Math. (Basel).--- 1984.--- V.53.--- P.397--399.
Pages	139-150
Volume	29
Issue	2
Year	2008
Journal	Matematychni Studii
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