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Testing commutativity of a group and the power of randomization

Published online by Cambridge University Press:  01 April 2012

Igor Pak*
Department of Mathematics, University of California, Los Angeles, 530 Portola Plazza 6363 Math Sciences Building Los Angeles, CA 90095-1555, USA (email:


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Let G be a group generated by k elements, G=〈g1,…,gk〉, with group operations (multiplication, inversion and comparison with identity) performed by a black box. We prove that one can test whether the group G is abelian at a cost of O(k) group operations. On the other hand, we show that a deterministic approach requires Ω(k2) group operations.

Research Article
Copyright © London Mathematical Society 2012


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