Geodesic
A short proof of the Khukhro-Makarenko theorem on large characteristic subgroups with laws
Sbornik. Mathematics, Tome 200 (2009) no. 5, pp. 661-664 Cet article a éte moissonné depuis la source Math-Net.Ru

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We give a short proof and a somewhat stronger version of the Khukhro-Makarenko theorem that each group which virtually satisfies an outer commutator identity contains a finite-index characteristic subgroup satisfying this identity. An estimate for the index of this characteristic subgroup is obtained. Bibliography: 3 titles.
Keywords: characteristic subgroups, outer commutator identities. 
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A. A. Klyachko; Yu. B. Mel'nikova. A short proof of the Khukhro-Makarenko theorem on large characteristic subgroups with laws. Sbornik. Mathematics, Tome 200 (2009) no. 5, pp. 661-664. http://geodesic.mathdoc.fr/item/SM_2009_200_5_a1/

[1] M. I. Kargapolov, Ju. I. Merzljakov, Fundamentals of the theory of groups, Grad. Texts in Math., 62, Springer-Verlag, New York–Berlin, 1979 | MR | MR | Zbl | Zbl

[2] E. I. Khukhro, N. Yu. Makarenko, “Large characteristic subgroups satisfying multilinear commutator identities”, J. Lond. Math. Soc. (2), 75:3 (2007), 635–646 | DOI | MR | Zbl

[3] N. Yu. Makarenko, E. I. Khukhro, “Large characteristic subgroups satisfying multilinear commutator identities”, Dokl. Math., 75:1 (2007), 112–114 | DOI | MR