# Bounds on the Number of Maximal Subgroups of Finite Groups: Applications
###### Abstract  
finite group  
and non-abelian chief factors (see Chapter 1 in [6]). The first invariant is related to primitive quotient groups of type 1. Definition 2: Let \(G\) be a group and let \(n>1\) be a natural number. We denote by \(\operatorname{cr}_{n}^{\mathfrak{M}}(G)\) the number of crowns associated to complemented abelian chief factors of order \(n\) of \(G\), that is, the number of \(G\)-isomorphism classes of complemented abelian chief factors of \(G\). Clearly, \(\operatorname{cr}_{n}^{\mathfrak{M}}(G)=0\) unless \(n\) is a power of a prime. The second invariant concerns non-abelian chief factors and is related to the primitive quotients of type 2. Definition 3: Let \(n\) be a natural number. The symbol \(\operatorname{rs}_{n}(G)\) denotes the number of non-abelian chief factors \(A\) in a given chief series of \(G\), such that the associated primitive group \(G/\mathsf{C}_{G}(A)\) has a core-free maximal subgroup of index \(n\). Our third and fourth invariants concern also non-abelian chief factors and contain information about the primitive quotients of type 3 of the group. Definition 4: Let \(n\) be a natural number. The symbol \(\operatorname{ro}_{n}(G)\) denotes the number of non-abelian chief factors \(A\) in a given chief series of \(G\), such that \(A\) has order \(n\). Definition 5: Let \(n\) be a natural number. The symbol \(\operatorname{rm}_{n}(G)\) denotes the maximum of the lengths of the \(G\)-crowns associated to non-abelian chief factors of order \(n\) of \(G\). Denote by \(\mathbb{T}\) the set of all prime powers greater than \(1\) and by \(\mathbb{S}\) the set of all powers greater than \(1\) of the orders of non-abelian simple groups. The main results of [4] are the following ones. Theorem 1 (Theorem B in [4]): The number \(\operatorname{m}_{n}(G)\) of maximal subgroups of index \(n\) of a \(d\)-generated group \(G\) satisfies the following bounds:  
\[\operatorname{m}_{n}(G) \leq(n^{d}-1)\operatorname{cr}_{n}^{\mathfrak{M}}(G)+n^{2} \operatorname{rs}_{n}(G) \text{if }n\in\mathbb{T},\] \[\operatorname{m}_{n}(G) \leq n^{2}\operatorname{rs}_{n}(G)+n^{2}\bigg{(}\frac{ \operatorname{rm}_{n}(G)\operatorname{ro}_{n}(G)}{2}\bigg{)}\leq n^{2} \operatorname{rs}_{n}(G)+n^{d+2}\bigg{(}\frac{\operatorname{ro}_{n}(G)}{2} \bigg{)} \text{if }n\in\mathbb{S},\] \[\operatorname{m}_{n}(G) \leq n^{2}\operatorname{rs}_{n}(G) \text{if }n\
otin\mathbb{S}\cup\mathbb{T}.\]  
Theorem 2: Let \(G\) be a \(d\)-generated non-trivial group. Then, for  
\[\eta(G):=\max\Bigl{\{}d+2.02+\max_{n\in\mathbb{T}}\{\log_{n}2+ \log_{n}\operatorname{cr}_{n}^{\mathfrak{M}}(G)\},\] \[\qquad\qquad\qquad\qquad 4.02+\max_{n\in\mathbb{S}}\bigl{\{}\log_{n} 2+\log_{n}\operatorname{rs}_{n}(G)\bigr{\}},\] \[\qquad\qquad\qquad\qquad 4.02+\max_{n\in\mathbb{S}}\bigl{\{}\log_{n} \operatorname{rm}_{n}(G)+\log_{n}\operatorname{ro}_{n}(G)\bigr{\}}\Bigr{\}},\]  
and  
\[\kappa(G):=\max\Bigl{\{}d+2.02+\max_{n\in\mathbb{T}}\{\log_{n}2+ \log_{n}\operatorname{cr}_{n}^{\mathfrak{M}}(G)\},\] \[\qquad\qquad\qquad\qquad 4.02+\max_{n}\{\log_{n}2+\log_{n} \operatorname{rs}_{n}(G)\},\] \[\qquad\qquad\qquad\qquad 4.02+d+\max_{n\in\mathbb{S}}\{\log_{n} \operatorname{ro}_{n}(G)\}\Bigr{\}},\]  
we find that  
\[\mathcal{V}(G)\leq\eta(G)\leq\kappa(G).\]Other bounds for \(\mathcal{V}(G)\) can be found in [7]. They depend on an invariant, defined there only for non-abelian characteristically simple groups, but that can be also defined for elementary abelian groups. Definition 6: Let \(G\) be a group. For a characteristically simple group \(A\), that is, a direct product of copies of a simple group \(S\), \(\mathsf{r}_{A}(G)\) denotes the largest number \(r\) such that \(G\) has a normal section that is the direct product of \(r\) non-Frattini chief factors of \(G\) that are isomorphic (not necessarily \(G\)-isomorphic) to \(A\). The main theorem of [8] gives a simpler reinterpretation of the invariant \(\mathsf{r}_{A}(G)\) as the number of non-Frattini chief factors isomorphic to \(A\) in a given chief series of \(G\).