The finite simple groups have complemented subgroup lattices. (Q1884532)

A group \(G\) is called a \(K\)-group if its subgroup lattice is a complemented lattice, i.e., for a given \(H\leq G\) there exists \(X\leq G\) such that \(\langle H,X\rangle=G\) and \(H\cap X=1\). In this note, the authors give a full answer to a long-standing open question in finite group theory, by proving that every finite simple group is a \(K\)-group. Previously, \textit{E. Previato} [Boll. Unione Mat. Ital., VI. Ser., B 1, 1003-1014 (1982)] showed that the simple groups \(A_n\), \(\text{PSL}_n(q)\) and \(Sz(q)\) are \(K\)-groups. The proof relies on the classification of finite simple groups and on some structural properties of the maximal subgroups in these groups.