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General case: character tables determining 2-generated Sylow p-subgroups

Determine whether, for every finite group G and prime p, the character table of G suffices to decide whether the Sylow p-subgroups of G are 2-generated. This seeks to extend results known for certain classes of groups to the full generality of all finite groups.

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Background

In the introduction, the paper notes prior work showing that for certain classes of groups, the character table determines whether Sylow p-subgroups are 2-generated. The authors emphasize that this general determination is not yet known beyond those classes.

Their main theorem establishes such a determination (via Galois action on principal blocks) for almost simple groups when p=3, suggesting progress toward a broader local/global characterization.

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