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We analyze the computational complexity of solving a single equation and checking identities over finite meta-abelian groups. Among others we answer a question of Goldmann and Russel from '98: We prove that it is decidable in polynomial time whether or not an equation over the six element group S 3 has a solution.doi:10.1142/s0218196706003256 fatcat:eo5ahy36ajaynioq7gbobb4mui