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Algorithm for deciding 2-generation of Sylow p-subgroups from character tables (p ≥ 3)

Develop an algorithm that, given the character table of a finite group G, decides whether the Sylow p-subgroups of G are 2-generated for primes p ≥ 3.

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Background

The paper provides an algorithmic criterion for almost simple groups at p=3 by counting σ-fixed height-zero characters in the principal 3-block. However, the authors explicitly remark that, in general, an algorithm for p ≥ 3 has not yet been determined.

Formulating such an algorithm would extend the local/global connections explored here and in related work, aiming for a practical decision procedure across all finite groups and primes p ≥ 3.

References

It was also shown by Moret o and Samble in that for certain classes of groups the character table determines whether or not a group has $2$-generated Sylow $p$-subgroups; however, the general case remains open and, further, an algorithm to determine this properties for $p \ge 3$ has yet to be determined.

— Characters and the Generation of Sylow 3-Subgroups For Almost Simple Groups (2509.02854 - Ketchum, 2 Sep 2025) in Introduction