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.

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