On non-projective block components of Lefschetz characters for sporadic geometries

This work examines the possible projectivity of 2-modular block parts of non-projective Lefschetz characters over 2-local geometries of several sporadic groups. Previously known results on $M_{12}$, $J_2$, and $HS$ are mentioned for completeness. The main new results are on the sporadic groups $Suz$, $Co_3$, $Ru$, $O'N$, and $He$. For each group, the Lefschetz character is calculated, and the 2-modular blocks are examined for projectivity. In each case it is confirmed that a non-principal block contains a non-projective summand. The case of $O'N$ is additionally found to have a non-projective summand in its principal block part.