Characterization of finite simple nonabelian groups via twisted sets. (Q610300)

A subset \(K\) of a group \(G\) is called `twisted' if \(1\in K\) and \(xy^{-1}x\in K\) for all \(x,y\in K\). The main theorem in this paper is the following: Let \(K\) and \(S\) be two twisted subsets of some groups \(G\) and \(L\), respectively, such that \(G\neq K\), \(G=\langle K\rangle\) and \(L\neq S\), \(L=\langle S\rangle\). Suppose that \(G\) is a finite simple nonabelian group. If \(K\) is isomorphic to \(S\) (as twisted subsets) then \(L\) is a perfect central extension of \(G\). The author also gives an example showing that the converse to this theorem is false, namely, an arbitrary perfect central extension of a finite simple nonabelian group \(G\) need not be generated by a twisted subset isomorphic to \(K\).