Let G be a finite group, let \(\pi\) be a set of primes and let S be a subset of G. A \(\pi\)-section of G is the set of all elements of G whose \(\pi\)-parts are conjugate in G. The union of all \(\pi\)-sections of G which intersect S is denoted by \(S^{G,\pi}\). If \(H\leq G\), \(H^*\) denotes the set \(H\setminus \{1\}\). If n is a natural number, the \(\pi\)- part of n is denoted by \(n_{\pi}.\)
The main result of this paper asserts that G has a normal \(\pi\)- complement iff the following conditions hold: a) G has a Hall \(\pi\)- subgroup H. b) For every \(x\in H\setminus Z(G)\) the number of \(\pi '\)- elements of \(C_ G(x)\) is \(| C_ G(x)|_{\pi '}\). c) \(| (H^*)^{H,\pi}| =| G:H| | H^*|\). The proof is elementary but depends on results of R. Brauer [Math. Z. 83, 72-84 (1964; Zbl 0116.018)] and H. S. Leonard [Proc. Am. Math. Soc. 95, 5-6 (1985; Zbl 0572.20012)]. Two corollaries are also given.