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# Hardness of Equations over Finite Solvable Groups Under the Exponential Time Hypothesis

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## Acknowledgements

I am grateful to Moses Ganardi for bringing my attention both to the AND-weakness conjecture and to the exponential time hypothesis. I am also thankful to David A. Mix Barrington for an interesting email exchange concerning the AND-weakness conjecture and the idea to include steps of the lower central series in Proposition 8 to get a more refined upper bound. Furthermore, I am indebted to Caroline Mattes and Jan Philipp Wächter for many helpful discussions. Finally, I want to thank the anonymous referees for their valuable comments.

## Cite As

Armin Weiß. Hardness of Equations over Finite Solvable Groups Under the Exponential Time Hypothesis. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 102:1-102:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ICALP.2020.102

## Abstract

Goldmann and Russell (2002) initiated the study of the complexity of the equation satisfiability problem in finite groups by showing that it is in 𝖯 for nilpotent groups while it is 𝖭𝖯-complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups of Fitting length two. In this work, we present the first lower bounds for the equation satisfiability problem in finite solvable groups: under the assumption of the exponential time hypothesis, we show that it cannot be in 𝖯 for any group of Fitting length at least four and for certain groups of Fitting length three. Moreover, the same hardness result applies to the equation identity problem.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Problems, reductions and completeness
##### Keywords
• equations in groups
• solvable groups
• exponential time hypothesis

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