Detecting laws in power subgroups

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 G^m 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 $W^m$ and $W^n$ 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.

Keywords:

• Engel conditions
• group identities
• group laws
• group presentations
• power subgroups
• residually nilpotent groups

2010 Mathematics Subject Classification:

• 20E10 (primary)
• 20F19
• 20F45
• 20F05 (secondary)

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.

Additional information

Funding

This work was partially supported by the Clarendon Fund, Balliol College Marvin Bower Scholarship, and the James Fairfax Oxford Australia Scholarship.