On the number of maximal subgroups of a finite solvable group.
Let \(G\) be a finite soluble group, \(p\) a prime, \(m(G)\) the number of maximal subgroups of \(G\), and \(m_p(G)\) the number of maximal subgroups of \(G\) whose index is a power of \(p\). \textit{M. Herzog} and \textit{O. Manz} [J. Aust. Math. Soc., Ser. A 58, No. 1, 134-141 (1995; Zbl 0831.20030)] and \textit{R. J. Cook}, \textit{J. Wiegold} and \textit{A. G. Williamson} [ibid. A 43, 220-223 (1987; Zbl 0625.20016)] have provided upper bounds for \(m(G)\) and \textit{G. E. Wall} [J. Aust. Math. Soc. 2, 35-59 (1961; Zbl 0121.27302)] for both \(m(G)\) and \(m_p(G)\). The following two results are improvements of these bounds. If \(p^k\) is the order of a Sylow \(p\)-subgroup of \(G\) and \([G:O^p(G)]= p^r\), then \[ m_p(G)\leq\min\Biggl({p^r-1\over p-1}+{p^{k-r+1}-p\over p-1},{p^{k+1}-p\over p-1}\Biggr), \] and if \(r>1\), then \(m_p(G)\leq{p^k-1\over p-1}\) (Theorem 3). If \(|G|=p^{r_1}_1p^{r_2}_2\cdots p^{r_m}_m\), where \(p_1,p_2,\dots,p_m\) are distinct primes, \(r_1,r_2,\dots,r_m\) are positive integers and \(p^{r_1}_1=\min\{p^{r_i}_i\mid 1\leq i\leq m\}\), then \[ m(G)\leq {p^{r_1}_1-1\over p_1-1}+\sum^m_{i=2}{p^{r_i+1}_i-p_i\over p_i-1} \] (Corollary 4). -- Moreover, a sufficient condition is provided for the order of \(G\) under which this bound is best possible.
scientific article; zbMATH DE number 5956229