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On Ditsman’s lemma

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Abstract

Let H be a subgroup of a group G generated by a finite G-invariant subset X = tU k i=1 C i that consists of elements of finite order, where C i is a class of conjugate elements of G with representative a i . We prove that

$|H| \leqslant \prod\limits_{i = 1}^k {o(a_i )^{|C_i |} } ,$

where o(a i ) is the order of the element a i ∈ C i . Best estimates are obtained for some important special cases.

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References

  1. V. A. Belonogov, A Book of Problems in Group Theory (Nauka, Moscow, 2000) [in Russian].

    Google Scholar 

  2. M. A. Grechkoseeva, “On minimal permutation representations of classical simple groups,” Sib. Math. J. 44(3), 443–462 (2003).

    Article  MathSciNet  Google Scholar 

  3. D. Gorenstein, Finite Simple Groups: An Introduction to Their Classification (Plenum, New York, 1982; Mir, Moscow, 1985).

    Book  MATH  Google Scholar 

  4. A. P. Ditsman, “On p-groups,” Dokl. Akad. Nauk SSSR 258, 71–76 (1937).

    Google Scholar 

  5. L. S. Kazarin, “Burnside’s pα-lemma,” Math. Notes 48(2), 749–751 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  6. A. G. Kurosh, The Theory of Groups (Nauka, Moscow, 1967) [in Russian].

    Google Scholar 

  7. V. D. Mazurov, “Minimal permutation representations of finite simple classical groups. Special linear, symplectic and unitary groups,” Algebra Logic 32(3), 142–153 (1993).

    Article  MathSciNet  Google Scholar 

  8. J. H. Conway, R. T. Curtis, S. P. Norton, et al., Atlas of Finite Groups (Clarendon, Oxford, 1985).

    MATH  Google Scholar 

  9. R. Brauer and P. Fong, “On the centralizers of p-elements in finite groups,” Bull. London Math. Soc. 6, 319–324 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Maróti, “On the orders of primitive groups,” J. Algebra 258(2), 631–640 (2002).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics, Demidov Yaroslavl State University, ul. Soyuznaya 144, Yaroslavl, 150008, Russia

    L. S. Kazarin

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  1. L. S. Kazarin
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Correspondence to L. S. Kazarin.

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Original Russian Text © L.S. Kazarin, 2013, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2013, Vol. 19, No. 3.

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Kazarin, L.S. On Ditsman’s lemma. Proc. Steklov Inst. Math. 285 (Suppl 1), 91–98 (2014). https://doi.org/10.1134/S0081543814050095

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  • DOI: https://doi.org/10.1134/S0081543814050095

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