Permutation groups whose non-identity elements have \(k\) fixed points (Q2705008)

Let \(K\) be a set of natural numbers. A permutation group \(G\) is said to have type \(K\) if the number \(\text{fix}(g)\) of fixed points of every non-identity element \(g\in G\) belongs to \(K\). This note mostly deals with groups of type \(\{k\}\). A classification of groups of type \(\{2\}\) is due to \textit{N. Iwahori} [J. Fac. Sci., Univ. Tokyo, Sect. I 11, 47-64 (1964; Zbl 0201.36503)].NEWLINENEWLINENEWLINEThe main result is Theorem 2: Let \(G\) be a finite permutation group of type \(\{k\}\). Then there exists a finite list \(L(k)\) of permutation groups such that either (a) \(G\) has a fixed set \(\Delta\) of size \(k\), and the kernel \(N\) of the action of \(G\) on \(\Delta\) is semiregular outside \(\Delta\); or (b) \(G\) belongs to \(L(k)\). -- Furthermore in case (a) \(N\) is nilpotent if \(G\) acts non-trivially on \(\Delta\); and \(N\) is Abelian if the constituent \(G^\Delta\) has even order.NEWLINENEWLINENEWLINEIn case of locally finite groups \(G\) of type \(\{k\}\) a similar result is established (Theorem 4; see also the remark added in proof, referring to a theorem of \textit{E. I. Khukhro} [Mat. Sb., Nov. Ser. 130(172), No. 1(5), 120-127 (1986; Zbl 0608.20025)]).NEWLINENEWLINENEWLINEThe following conjecture is proposed in addition: Let \(G\) be a permutation group having type \(K\) where \(|K|>1\). Then either \(G\) has a base of cardinality \(|K|\), or \(G\) is one of a finite list \(L(K)\) of permutation groups.