On a problem of Sergienko (Q1802153)

This note contains a proof of the following conjecture of \textit{V. I. Sergienko}, which appears in Dokl. Akad. Nauk BSSR. 25, 200-203 (1981; Zbl 0468.20016) and also in The Kourovka Notebook [Transl., II. Ser., Am. Math. Soc. 121 (1983; Zbl 0512.20001), Problem 7.44]: If \(G\) is a finite group, if \(H\triangleleft G\) and every Sylow \(p\)-subgroup of \(H\) is a direct factor of some Sylow \(p\)-subgroup of \(G\) for all primes \(p\in \pi(G)\), then \(H\) has a complement in \(G\). The proof is short and relies at a key point on a result of \textit{F. Gross} [J. Algebra 77, 202-233 (1982; Zbl 0489.20019)]; it is shown first that if \(G\) is a counterexample of minimal order, then \(H\) must be solvable. But in this case the minimal counterexample does not exist by a result of Sergienko [loc.cit.].