Fitting-free group isomorphism is in AC^3 with multiplication tables and GI-hard with permutation generators, but the paper's FO O(log log n)-variable identification proof uses a false lemma.
Polynomial-time isomorphism test for groups with no abelian normal subgroups - (extended abstract)
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On the Complexity of Identifying Groups without Abelian Normal Subgroups: Parallel, First Order, and GI-Hardness
Fitting-free group isomorphism is in AC^3 with multiplication tables and GI-hard with permutation generators, but the paper's FO O(log log n)-variable identification proof uses a false lemma.