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[PLEASE SEE COMMENT] We consider the isomorphism problem for finite abeliangroups and finite meta-cyclic groups. We prove that for a dense set of positiveintegers $n$, isomorphism testing for abelian groups of black-box type of order$n$ can be done in time polynomial in $\log n$. We also prove that for a denseset of orders $n$ with given prime factors, one can test isomorphism forcoprime meta-cyclic groups of black-box type of order $n$ in time polynomial in$\log n$. Prior methods for these two classes of groups have running timesexponential in $\log n$.

Polynomial-time isomorphism testing of groups of most finite orders