On the maximum number of fixed points in automorphisms of prime order of 2-\((v,k,\text 1)\) designs (Q1387612)

2-\((v,k,1)\) designs with automorphisms of prime order \(p\), having the maximum possible number of fixed points are studied. An upper bound is obtained on the number of fixed points. The structure of designs in which the bound is met with equality (such a design is called \(p\text{-MFP} (v,k))\) is studied. Necessary and sufficient conditions on \(v\) for the existence of a \(p\text{-MFP} (v,k)\) for \(3\leq k\leq 5\) and any prime \(p\equiv 1\bmod k(k-1)\) are established.