# Bounds on the Number of Maximal Subgroups of Finite Groups: Applications
###### Abstract
The following result was proved in [7]. We present here a corrected version available in [9] due to a misprint in the originally published version. Theorem 3.1 (Theorem 9.5 in [9]): _Let \(G\) be a \(d\)-generated group. Then,_  
\[\operatorname{m}_{n}(G) \leq n^{2}\operatorname{rs}_{n}(G)+n^{2}\bigg{(}\frac{ \operatorname{rm}_{n}(G)\operatorname{ro}_{n}(G)}{2}\bigg{)}\leq n^{2} \operatorname{rs}_{n}(G)+n^{d+2}\bigg{(}\frac{\operatorname{ro}_{n}(G)}{2} \bigg{)} \text{if }n\in\mathbb{S},\]0  
_where \(c\) and \(c_{7}\) are two absolute constants._  
Here, the symbol \(\mathrm{rk}_{n}(G)\) denotes the maximum of the numbers \(\mathsf{r}_{A}(G)\), where \(A\) runs over the non-abelian characteristically simple groups \(A\) with \(1(A)\leq n\), where \(1(X)\) denotes the least degree of a faithful transitive permutation representation of a group \(X\), that is, the smallest index of a core-free subgroup of \(X\). The lower bound in Theorem 3.1 depends on the following result. Lemma 1 (Corollary 9.3 in [9]): _Let \(G\) be a group. Then, \(\mathsf{m}_{x}(G)\geq\mathrm{rk}_{n}(G)/n^{c_{7}}\) for some \(x\leq n^{c_{7}}\)._  
Our aim in this paper is twofold. In the first place, we establish significant improvements of the lower bound of Theorem 3.1. We prove:  
Theorem 3.2: _Let \(G\) be a \(d\)-generated non-trivial group. Then,_  
\[\operatorname{m}_{n}(G) \leq n^{2}\operatorname{rs}_{n}(G)+n^{2}\bigg{(}\frac{ \operatorname{rm}_{n}(G)\operatorname{ro}_{n}(G)}{2}\bigg{)}\leq n^{2} \operatorname{rs}_{n}(G)+n^{d+2}\bigg{(}\frac{\operatorname{ro}_{n}(G)}{2} \bigg{)} \text{if }n\in\mathbb{S},\]1  
We obtain our lower bound by means of an improved version of Lemma 1. This is done in two ways. On the one hand, we show that the constant \(c_{7}\) can be taken to be \(2\) and, on the other hand, we obtain a larger lower bound for the number of maximal subgroups of index \(x\). Moreover, if we impose some restrictions to the composition factors of the group, the bounds are further improved (see Theorem 3.1). Theorem 3.3: _Let \(G\) be a group. Then, \(\mathsf{m}_{x}(G)\geq x\lceil(2/3)\mathrm{rk}_{n}(G)\rceil\) for some \(x\leq n^{2}\)._  
Here, the symbol \(\lceil x\rceil\) denotes the excess integer part of \(x\), that is, the smallest integer number \(n\), such that \(x\leq n\). The symbol \(\lfloor x\rfloor\) will denote the defect integer part of \(x\), that is, the largest integer number \(n\), such that \(n\leq x\). The proof of Theorem 3.3 will be presented in Section 2. It depends on some results proved in [10] about maximal subgroups of small index in almost finite groups. In the second place, we compare the bounds of Theorem 1 with the ones of Theorem 3.1. This is only possible if we precise the values of the constants appearing in Theorem 3.1. As far as we know, the values of the constants \(c\) and \(c_{7}\) in Theorem 3.1 have not been estimated in the literature. We obtain, in Section 3, the value of the constant \(c\) of Theorem 3.1 and we prove the following result. **Theorem 6**. : _Let \(G\) be a \(d\)-generated group. Then,_  
\[\operatorname{m}_{n}(G) \leq n^{2}\operatorname{rs}_{n}(G)+n^{2}\bigg{(}\frac{ \operatorname{rm}_{n}(G)\operatorname{ro}_{n}(G)}{2}\bigg{)}\leq n^{2} \operatorname{rs}_{n}(G)+n^{d+2}\bigg{(}\frac{\operatorname{ro}_{n}(G)}{2} \bigg{)} \text{if }n\in\mathbb{S},\]2  
_where \(c=375.06\)._  
A slight variation of the proof of Theorem 6 gives smaller values for some of the constants (see Theorem 20). We do it in Section 3. The following example, that appears in Example 3.4 in [4], depends on a couple of constructions of subdirect products also introduced in that paper. In this example, we see that our bound for \(\mathcal{V}(G)\) improves dramatically the one of [7]. **Example 1**. : _There are three isomorphism classes of \(2\)-generated primitive groups of type 1 with socle of order \(8\), namely \(G_{1}=[\mathrm{C}_{2}^{3}]C_{7}\), \(G_{2}=[\mathrm{C}_{2}^{3}][C_{7}]C_{3}\) and \(G_{3}=[\mathrm{C}_{2}^{3}]\mathrm{GL}_{3}(2)\).