Let \(G\) be a finite group. A sequence \( (g_1, \dots, g_n)\) of elements of \(G\) is said to be irredundant if \(\langle g_j \mid j \neq i \rangle\) is properly contained in \( \langle g_1, \dots, g_n \rangle\) for every \(i \in \{ 1, \dots, n \}\). Let \(i(G)\) be the maximum size of any irredundant sequence in \(G\) and let \(m(G)\) be the maximum size of any irredundant generating sequence \( \langle g_1, \dots, g_n \rangle\) (i.e. an irredundant sequence with the property that \(G= \langle g_1, \dots, g_n \rangle\)). Over the last years several authors have analized this invariant. And recently, the natural connection between irredundant generating sequences and certain configurations of maximal subgroups of a group has been investigated. A family \(\{H_i \mid i \in I\}\) of subgroups of \(G\) is said to be in general position if, for every \(i \in I\), the intersection \(\bigcap_{i\neq j}H_{j}\) properly contains \(\bigcap_{j \in I}H_{j}\). The size of the largest family of maximal subgroups of \(G\) in general postion is denoted by \(\operatorname{MaxDim}(G)\).
It is known that the difference \(\operatorname{MaxDim}(G)-m(G)\) can be arbitrarily large. In the paper under review, this difference is analized in case \(G\) is a finite soluble group. The main results are the following:
Theorem 1. If \(G\) is a finite group and the derived subgroup \(G'\) is nilpotent, then \(\operatorname{MaxDim}(G)=m(G)\).
Theorem 2. For any odd prime \(p\), there exists a finite group \(G\) with Fitting length two such that \(m(G)=3\), \(\operatorname{MaxDim}(G)=p\) and \(i(G)=2p\).
It is also asserted that \(G'\) is nilpotent when \(m(G) \leq 2\). So, if \(G\) is a soluble group with \(\operatorname{MaxDim}(G) \neq m(G)\), then \(m(G) \geq 3\).