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Mikls Abrt

On the probability of satisfying a word in a group

We show that for any finite group G and for any d there exists a word w ∈ F_d such that a d-tuple in G satisfies w if and only if it generates a solvable subgroup. As a corollary, the probability that a word is satisfied in a fixed non-solvable group can be made arbitrarily small; this answers a question of Alon Amit.

It also follows that there is no absolute bound in the Baumslag–Pride theorem for the minimal index in a group with at least two more generators than relators of a subgroup that can be mapped homomorphically onto a non-abelian free group.

Journal of Group Theory, Walter de Gruyter

Print ISSN: 1433-5883
Volume: 9, 09/2006
Pages: 685 - 694

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