Abstract
Every finite non-nilpotent group can be extended by a term operation such that solving equations in the resulting algebra is NP-complete and checking identities is co-NP-complete. This result was firstly proven by Horváth and Szabó; the term constructed in their proof depends on the underlying group. In this paper we provide a uniform term extension that induces hard problems. In doing so we also characterize a big class of solvable, non-nilpotent groups for which extending by the commutator operation suffices.
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Aichinger, E., Mudrinski, N., Opršal, J.: Complexity of term representations of finitary functions. Int. J. Algebra Comput. 28, 1101–1118 (2018)
Baer, R.: Engelsche Elemente Noetherscher Gruppen. Math. Ann. 133, 256–270 (1957)
D. M. Barrington, P. McKenzie, C. Moore, P. Tesson, and D. Thérien, Equation satisfiability and program satisfiability for finite monoids, in: International Symposium on Mathematical Foundations of Computer Science (Bratislava, 2000), Lecture Notes in Comput. Sci., 1893, Springer (Berlin, 2000), pp. 172–181
Burris, S., Lawrence, J.: The equivalence problem for finite rings. J. Symbolic Comput. 15, 67–71 (1993)
Burris, S., Lawrence, J.: Results on the equivalence problem for finite groups. Algebra Universalis 52, 495–500 (2005)
Földvári, A.: The complexity of the equation solvability problem over semipattern groups. Int. J. Algebra Comput. 27, 259–272 (2017)
Földvári, A.: The complexity of the equation solvability problem over nilpotent groups. J. Algebra 495, 289–303 (2018)
Goldmann, M., Russell, A.: The complexity of solving equations over finite groups. Inform. and Comput. 178, 253–262 (2002)
Gorazd, T., Krzaczkowski, J.: The complexity of problems connected with two-element algebras. Rep. Math. Logic 46, 91–108 (2011)
G. Horváth, Functions and Polynomials over Finite Groups from the Computational Perspective, PhD thesis, University of Hertfordshire (2008)
Horváth, G.: The complexity of the equivalence and equation solvability problems over nilpotent rings and groups. Algebra Universalis 66, 391–403 (2011)
G. Horváth, J. Lawrence, and R. Willard, The complexity of the equation solvability problem over finite rings, preprint, http://real.mtak.hu/28210/ (2015)
Horváth, G., Mérai, L., Szabó, C., Lawrence, J.: The complexity of the equivalence problem for nonsolvable groups. Bull. London Math. Soc. 39, 433–438 (2007)
Horváth, G., Szabó, C.: The extended equivalence and equation solvability problems for groups. Discrete Math. Theor. Comput. Sci. 13, 23–32 (2011)
Horváth, G., Szabó, C.: Equivalence and equation solvability problems for the alternating group \(A_4\). J. Pure Appl. Algebra 216, 2170–2176 (2012)
Hunt III, H.B., Stearns, R.E.: The complexity of equivalence for commutative rings. J. Symbolic Comput. 10, 411–436 (1990)
P. M. Idziak and J. Krzaczkowski, Satisfiability in multi-valued circuits, in: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, ACM (2018), pp. 550–558
Kompatscher, M.: The equation solvability problem over supernilpotent algebras with Mal'cev term. Int. J, Algebra Comput (2018)
D.-J. Robinson, A Course in the Theory of Groups, Springer (Berlin–Heidelberg, 1996)
Acknowledgement
The author would like to thank Attila Földvári and Gábor Horváth for their valuable feedback on earlier versions of this paper.
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This work has been supported by Charles University Research Centre programs No. PRIMUS/SCI/12 and No. UNCE/SCI/022, as well as grant 18-20123S of the Czech Grant Agency (GAČR).
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Kompatscher, M. Notes on extended equation solvability and identity checking for groups. Acta Math. Hungar. 159, 246–256 (2019). https://doi.org/10.1007/s10474-019-00924-7
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DOI: https://doi.org/10.1007/s10474-019-00924-7