Finite \(p\)-groups with small subgroups generated by two conjugate elements.
A \(p\)-group \(G\) is said to be a \(K_1\)-group if \(|\langle x,x^y\rangle:\langle x\rangle|\leq p\) for all \(x,y\in G\). The main result is the following: Theorem. Let a \(p\)-group \(G\) be a \(K_1\)-group. (a) If \(p\geq 3\), then \(\mho_1(G)\leq Z(G)\) and \(\exp(G')=p\). If, in addition, \(p=3\), then \(\mathrm{cl}(G)\leq 4\). If \(p\geq 5\), then \(\mathrm{cl}(G)\leq 3\). (b) If \(p=2\), then \(\mho_2(G)\leq\text Z(G)\), \(\exp(G')\leq4\) and \(\mathrm{cl}(G)\leq 3\). Also the following proper subclass \(K_2\) of the class \(K_1\) is considered. A \(p\)-group \(G\) is said to be a \(K_2\)-group if \(|\langle H,H^x \rangle:\langle H\rangle|\leq p\) for all \(x\in G\) and \(H\leq G\). If \(G\) is a \(K_2\)-group of exponent \(p\), then \(|G'|\leq p\). In general case, \(\mathrm{cl}(G)\leq 3\), and if \(p\geq 3\), then \(G'\) is elementary Abelian.
scientific article; zbMATH DE number 6161243