Replacement theorem by characteristic subgroup satisfying a multilinear commutator identity

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This article defines a replacement theorem
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Statement

Suppose G is a group and H is a subgroup satisfying a multilinear commutator identity: in other words, there is a word w(x_1,x_2,\ldots,x_n) composed entirely of iterated commutation, such that w(x_1,x_2,\ldots,x_n) = e for all x_i \in H. Then, there exists a characteristic subgroup of finite index N of G satisfying the same multilinear commutator identity, and where the index of N is bounded from above by a function of n and w (independent of G).

Particular cases

  • If H is nilpotent of class c, we can find a characteristic subgroup of finite index, also nilpotent of the same class c, and with index bounded from above by a function of n and c.
  • If H is solvable with derived length l, we can find a characteristic subgroup of finite index, also solvable with derived length l, and with index bounded from above by a function of n and l.

References

  • Large characteristic subgroups satisfying multilinear commutator identities by Evgenii I. Khukhro and N. Yu. Makarenko, Journal of the London Mathematical Society, July 2007, Page 1-12: This proves that if a group has a subgroup of finite index satisfying a multilinear commutator identity, then we can find a characteristic subgroup of finite index satisfying the same identity. The result is used to prove facts about fixed point-free automorphisms on nilpotent groupsMore info