Minimal Systems of Generators and Relations and Properties of Wreath Products of Perfect Groups

Автори

Generators and defining relations for wreath products of perfect group which is two generating and alternating groups (m  2 times) are given. System of generators of metaperfect groups are found. Generators and defining relations for wreath products of 2-generating perfect groups were found, including alternating groups, i.e. (m  2 time). Systems of generators for metaperfect groups were investigated. A constructive proof of the minimality found system of generators was presented. It is shown that metaperfect group is not locally finite group. Cases of wreath product of metaperfect group with group which may be such that acts as a transitive and intransitive, construct the corresponding systems of generators. Presented generalization is the appearance of the product of different perfect groups and finding the exact value instead of the estimate. As it was found, for perfect 2-generator groups, wich has conditions founded by us, satisfy the equality of lower estimate it was easily generalized for 3-generated groups as It was found, that some of the properties immanent alternating groups are saved for metaalternating groups. The criterion of perfectly for metaalternating group was obtained. Inverse limit of metaalternating group, which іs proved to be branch group that does not own the property of locally finiteness was analyzed.

Publication year: 
2014
Issue: 
4
УДК: 
582.284.3
С. 93–101., Бібліогр.: 17 назв.
References: 

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References [transliteration]: 

1. M. Bhattacharjee, “The probability of generating certain profinite groups by two elements”, Israel J. Math., no. 86, pp. 311–329, 1994.
2. M. Quick, “Probabilistic generation of wreath products of non-Abelian finite simple groups”, Commun. Algebra, no. 32 (12), pp. 4753–4768, 2004.
3. M. Quick, “Probabilistic generation of wreath products of non-Abelian finite simple groups. II”, Int. J. Algebra Comput., no. 16(3), pp. 493–503, 2006.
4. Zavodi͡a M.V., Sikora V.S., Sushchans′kyĭ V.I. Dvukhelementni systemy tvirnykh metaznakozminnykh hrup skinchennoho ranhu // Mat. ctudiï. – 2010. – 34, # 1. – S. 3–12.
5. Oliĭnyk B.V., Sikora V.S., Sushchans′kyĭ V.I. Metasymmetrycheskye y metaznakoperemennыe hruppы beskonechnoho ranha // Mat. studiï. – 2008. – 29, # 2. – S. 139–150.
6. I.V. Bondarenko, “Finite generation of iterated wreath products”, Archiv der Mathematik, vol. 95, is. 4, pp 301–308, 2010.
7. R.D. Karmichael, “Abstract definitions of the symmetric and alternating groups and certain other permutation groups”, Quart. J. Math., vol. 49, pp. 226–270, 1923.
8. Sikora V.S., Sushchans′kyĭ V.I. Operat͡siï na hrupakh pidstanovok. – Chernivt͡si: Ruta, 2003. – 256 s.
9. Sushchanskiĭ V.I. Normal'noe stroenie gruppy izometriĭ metricheskogo prostranstva t͡selykh p-adicheskikh chisel. Algebraicheskie struktury i ikh primenenie. – K.: KGU, 1988 – S. 113–121.
10. R. Grigorchuk, I. Pak, Groups of Intermediate Growth: an Introduction for Beginners. Preprint [Online]: http://arxiv.org/pdf/math/0607384.pdf
11. W.M. Kantor, “Some Сayley graphs for simple groups”, Discrete Applies Mathematics, vol. 25, pp. 99–104, 1989.
12. H. Wielandt, Finite permutations Groups. New York–London: Academic Press, 1968. – 108 p.
13. D. Segal, “The finite images of finitely generated groups”, Proc. Lond. Math. Soc., III. Ser., vol. 82(3), pp. 597–613, 2001.
14. J. Wiegold, “Growth sequences of finite groups 3”, J. Austral. Math. Soc., vol. 25, pp. 142–144, 1978.
15. Kaluzhnin L.A. Ob odnom obobshchenii silovskikh r-podgrupp simmetrit͡seskikh grupp // Acta Math. Hung. – 1951.– 2, # 3-4. – Р. 198–221.
16. R. Grigorchuk, Z. Sunic, “Self-similarity and branching in group theory”, Math. Res. Notices, p. 54, 1998.
17. R. Grigorchuk, Z. Sunic, Branch groups Laurent Bartholdi, 2005, p. 112.

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