An answer to two questions of Brewster and Yeh on \(M\)-groups. (Q1409757)

\textit{B. Brewster} and \textit{G. Yeh} [in J. Algebra 146, No. 1, 18-29 (1992; Zbl 0752.20005)], raised the following two questions: Question 1. Let \(A\) and \(B\) be normal subgroups of coprime indices in a finite group \(G\) such that \(A\) and \(B\) are \(M\)-groups. Is it true that \(G\) is an \(M\)-group? Question 2. Let \(A\) and \(B\) be normal subgroups of coprime orders in a finite group \(G\) such that \(G/A\) and \(G/B\) are \(M\)-groups. Is it true that \(G\) is an \(M\)-group? The author produces two examples giving negative answers of Questions 1 and 2. The following theorems disprove Questions 1 and 2, respectively. Theorem 2.2. Let \(p\) and \(q\) be odd primes satisfying \(q\equiv-1\pmod p\). Then there exists a group \(\Gamma=A\cdot E\), a semidirect product with kernel \(E\), where \(E\) is extraspecial of order \(q^5\) and \(A=\langle a\rangle\) is cyclic of order \(2p\), and this group is not an \(M\)-group but its subgroups \(\langle a^2\rangle\cdot E\) and \(\langle a^p\rangle\cdot E\) of coprime indices are \(M\)-groups. Theorem 2.4. Let \(p\) and \(q\) be as in the previous theorem. There exists a \(\{p,q\}\)-group \(G\) that is not an \(M\)-group such that \(G/O_p(G)\) and \(G/O_q(G)\) are \(M\)-groups. It is noticed, that the example which proves Theorem 2.4, was built by E. C. Dade for another aim.