On finite \(p\)-groups with metacyclic commutator subgroup (Q1288081)

For a finite \(p\)-group \(G\), denote by \(d(G)\) the minimal number of generators of \(G\). By \(G'\) denote the commutator subgroup of \(G\). The author proves the following assertions. Theorem 1. Let \(G\) be a finite 2-group such that \(d(G)\leq 2\) and the commutator subgroup of any 2-generated subgroup of \(G\) is cyclic. Then \(G''=1\). -- This theorem generalizes \textit{J. L. Alperin}'s theorem asserting that if \(p\) is an odd prime and \(G\) is a finite \(p\)-group in which the commutator subgroup of every 2-generated group is cyclic then \(G''=1\) [Trans. Am. Math. Soc. 106, 77-99 (1963; Zbl 0111.02803)]. The proof of Theorem 1 is based on the following assertions. Theorem 2. If the commutator subgroup of every 3-generated subgroup in a finite 2-group \(G\) is abelian and \(d(G)\leq 5\) then \(G''=1\). Theorem 3. Let \(G\) be a finite \(p\)-group whose commutator subgroup of any proper subgroup is an abelian group and \(d(G)\leq 2\). Then \(| G''|\leq 2\). The author presents some examples showing that the conditions of the theorems are essential.