Finite \(p\)-groups with many minimal nonabelian subgroups. (Q1758436)

The author in a series of papers, [such as Glas. Mat., III. Ser. 46, No. 2, 351-356 (2011; Zbl 1244.20014)], investigates the structure of finite \(p\)-groups. Let \(G\) be a nonabelian not minimal nonabelian finite \(p\)-group. Consider the following two properties: (*) any nonabelian subgroup \(H\) of \(G\) with an Abelian maximal subgroup is minimal nonabelian; (**) whenever \(X\) and \(Y\) are two minimal nonabelian subgroups, the elements \(x\in X\setminus Y\) and \(y\in Y\setminus X\) generate a minimal nonabelian subgroup. The author describes completely the groups \(G\) satisfying one of these properties. As to the latter one, if it is satisfied then the former one is also satisfied, \(p=2\), \(G\) is special of order 64 and isomorphic to the 2-Sylow subgroup of the simple group \(Sz(8)\). This answers problem 2331 in the forthcoming book by \textit{Y. Berkovich} and the author [Groups of prime power order. Vol. 4. Berlin: Walter de Gruyter (in preparation)].