The modular isomorphism problem asks whether a finite -group is determined, up to group isomorphism, by its group algebra over the finite field . The general assertion is false in characteristic 2, while the odd-characteristic problem remains open (Margolis 2022).

Tuzzolo (2026, Corollary 1.1) proves a positive answer for odd primes when the group has exponent and nilpotency class less than . Thus, for such a finite -group , an isomorphism with a finite -group forces . The result is a restricted positive case and does not settle the remaining odd-characteristic problem.

See also

Group Algebra

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References

Margolis, L. "The Modular Isomorphism Problem: A Survey." Jahresber. Dtsch. Math.-Ver. 124, 157-196, 2022. https://doi.org/10.1365/s13291-022-00249-5.Tuzzolo, D. "Short Jennings Series and the Modular Isomorphism Problem." 29 Oct 2026. https://arxiv.org/abs/2610.40868v1.

Cite this as:

Weisstein, Eric W. "Modular Isomorphism Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ModularIsomorphismProblem.html

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