Smooth groups (Q5933878)

A group is called smooth if it has a finite maximal chain of subgroups in which any two intervals of the same length are isomorphic (as lattices). The author shows that every finite smooth group \(G\) is a semidirect product of a \(p\)-group \(P\) by a cyclic group; and if \(G\) is neither cyclic nor a \(p\)-group, then \(P\) is Abelian of exponent at most \(p^2\) or extraspecial. In particular, a finite smooth group is solvable. The exact structure of \(G\) is determined if \(G\) is not a \(p\)-group. The proofs of these results are divided into many different cases.