On the existence of solvable normal subgroups in finite groups
Let \(G\) be a finite group and \(p\) a prime dividing the order of \(G\). If (i) \(G\) is \(p\)-soluble, (ii) \(x\) is a \(p\)-element of \(G\), and (iii) \(\langle x,x^g\rangle\) is a group of odd order for every \(g\in G\), then \(x\in O(G)\) (Theorem 1). If (i) all Sylow \(p\)-subgroups of \(G\) are Abelian, (ii) \(x,y\) are non-identity elements of \(G\), (iii) \(\langle x^g,y^h\rangle\) is a \(p\)-group for all elements \(g,h\in G\), and (iv) \(\langle x^G\rangle=\langle y^G\rangle\), then \(\langle x,x^g\rangle\subseteq O_p(G)\) (Theorem 2).
scientific article; zbMATH DE number 1238052