Abelian \(ABA\)-factorizations of finite groups (Q1377970)

Most finite groups are \(ABA\)-groups. Which groups \(G\), however, have an \(ABA\)-factorization with abelian proper subgroups \(A\) and \(B\)? If \(G\) is simple it must be \(\text{SL}(2,2^n)\) with \(| A|=2^n+1\) and \(| B|=2^n\) (Theorem 1). A finite \(ABA\)-group with abelian \(A\) and cyclic \(B\) is solvable (Theorem 2). A survey of the proof is included. It uses the classification of finite simple groups and relies on the estimates \(| B|<| G|^{1/2}\) and \(| A|^2| B|\geq| G|\) for abelian \(A\), \(B\) in \(ABA\)-groups with a unique Fitting subgroup; these estimates follow from a result of \textit{V. I. Zenkov} [Mat. Sb. 184, No. 6, 151-159 (1993; Zbl 0839.20031)].