Polynomial-time isomorphism test for solvable groups with abelian Sylow subgroups

Abstract

The group isomorphism problem in computational complexity asks whether two finite groups given by their Cayley tables are isomorphic or not. Although polynomial-time isomorphism tests exist for many specific types of groups, no general polynomial-time algorithm is known, classes of solvable and nilpotent groups being the main obstacles. In 2012 Babai and Qiao gave a polynomial-time isomorphism test for the class of solvable groups admitting normal series with abelian Sylow factors. We generalize their result and give a polynomial-time isomorphism test for solvable A-groups, i.e. solvable groups with abelian Sylow subgroups. The algorithm heavily relies both on the computational methods developed by Babai and Qiao, and structural properties of A-groups.

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BibTeX RIS

BibTeX

@misc{skresanov2026polynomialtime,
  title = {{Polynomial-time isomorphism test for solvable groups with abelian Sylow subgroups}},
  author = {Saveliy V. Skresanov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.26748},
  eprint = {2605.26748},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.26748},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Polynomial-time isomorphism test for solvable groups with abelian Sylow subgroups
AU  - Saveliy V. Skresanov
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2605.26748
UR  - https://arxiv.org/abs/2605.26748
AB  - The group isomorphism problem in computational complexity asks whether two finite groups given by their Cayley tables are isomorphic or not. Although polynomial-time isomorphism tests exist for many specific types of groups, no general polynomial-time algorithm is known, classes of solvable and nilpotent groups being the main obstacles. In 2012 Babai and Qiao gave a polynomial-time isomorphism test for the class of solvable groups admitting normal series with abelian Sylow factors. We generalize their result and give a polynomial-time isomorphism test for solvable A-groups, i.e. solvable groups with abelian Sylow subgroups. The algorithm heavily relies both on the computational methods developed by Babai and Qiao, and structural properties of A-groups.
ER  -

Abstract

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