# Replacement theorem by characteristic subgroup satisfying a multilinear commutator identity

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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