The fixity of permutation groups (Q1895606)

By definition, the fixity of a finite permutation group \(G\) is the maximal number of fixed points of a non-identity element of \(G\); so if \(f\) denotes the fixity of \(G\) and \(n\) the degree of \(G\) then \(n-f\) is the minimal degree of \(G\). The authors prove some general theorems on transitive permutation groups \(G\) with given fixity \(f>0\) where \(G\) is supposed in addition to be nilpotent (Theorem 1.1), solvable (Theorem 1.2) or primitive (Theorem 1.3). If \(G\) is nilpotent then \(G\) has a subgroup whose index and nilpotency class are both bounded by a function of \(f\). If \(G\) is primitive then either \(G\) has a solvable subgroup of derived length at most 4 whose index is bounded by a function of \(f\) or \(G\) is almost simple with \(F^*(G)\) similar to \(\text{PSL}(2,q)\) or \(Sz(q)\) in its natural doubly-transitive representation. Of course, the last-mentioned deep theorem depends on the classification of finite simple groups.