Total 3-closedness of PSL_n(2) for n greater than or equal to 5

Determine whether the finite simple groups PSL_n(2) are totally 3-closed for each integer n greater than or equal to 5, thereby completing the classification of total 3-closedness within the family of projective special linear groups.

Background

The paper classifies total 3-closedness for PSL_2(q) and PSL_3(q), proves that PSL_4(q) is never totally 3-closed, and proves that PSL_n(q) is not totally 3-closed when n is at least 5 and q is greater than 2. These results leave precisely the infinite family PSL_n(2) with n at least 5 unresolved.

The unresolved cases are not addressed by the semilinear obstruction, which requires a proper prime-power field extension or a nontrivial index between PGL_n(q) and PSL_n(q), nor by the central-vector obstruction, which requires q greater than 2. Resolving these cases would complete the classification for all finite nonabelian simple projective special linear groups.

References

Thus, within the family \PSL_n(q), the classification of total 3-closedness remains open only when n\geq 5 and q=2.

— Total 3-closure for projective special linear groups  (2608.17878 - Gong et al., 18 Aug 2026) in Remark following Theorem 1.1 in Section 1 (Introduction); reiterated in Section 5