Abstract
A group law is said to be detectable in power subgroups if, for all coprime m and n, a group G satisfies the law if and only if the power subgroups Gm and Gn both satisfy the law. We prove that for all positive integers c, nilpotency of class at most c is detectable in power subgroups, as is the k-Engel law for k at most 4. In contrast, detectability in power subgroups fails for solvability of given derived length: for all coprime m and n we construct a finite group W such that and
are metabelian but W has derived length 3. We analyse the complexity of the detectability of commutativity in power subgroups, in terms of finite presentations that encode a proof of the result.
2010 Mathematics Subject Classification:
Acknowledgements
I thank my supervisor Martin Bridson for guidance, encouragement, and helpful suggestions; Nikolay Nikolov for suggesting the problem of detectability of Engel laws; Peter Neumann for helpful conversations; Geetha Venkataraman for helpful correspondence; Robert Kropholler for helpful comments on a draft; my officemates L. Alexander Betts and Claudio Llosa Isenrich for helpful conversations; Yves Cornulier and Mark Sapir for their input on MathOverflow; and Alexander Olshanskii for suggesting the construction for Theorem B, which is more elegant and general that my original construction, and suggesting to extend Theorem A to the locally residually nilpotent case.
Disclosure statement
No potential conflict of interest was reported by the author.