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 is a group and is a subgroup satisfying a multilinear commutator identity: in other words, there is a word composed entirely of iterated commutation, such that for all . Then, there exists a characteristic subgroup of finite index of satisfying the same multilinear commutator identity, and where the index of is bounded from above by a function of and (independent of ).
Particular cases
- If is nilpotent of class , we can find a characteristic subgroup of finite index, also nilpotent of the same class , and with index bounded from above by a function of and .
- If is solvable with derived length , we can find a characteristic subgroup of finite index, also solvable with derived length , and with index bounded from above by a function of and .
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 groups^{More info}