On the structure of $$\mathfrak {N}_{\sigma }$$-critical groups

Let G be a finite group, \(\sigma =\{\sigma _{i}|i\in I\}\) be a partition of the set of all primes \(\mathbb {P}\) and \(\sigma (G)=\{\sigma _i|\sigma _i\cap \pi (G)\ne \emptyset \}\). G is said to be \(\sigma \)-primary if \(|\sigma (G)|\le 1\); \(\sigma \)-soluble if every chief factor of G is \(\sigma \)-primary. A set \(\mathcal {H}\) of subgroups of G is said to be a complete Hall \(\sigma \)-set of G if every non-identity member of \(\mathcal {H}\) is a Hall \(\sigma _i\)-subgroup of G for some \(\sigma _i\) and \(\mathcal {H}\) contains exactly one Hall \(\sigma _i\)-subgroup for every \(\sigma _i\in \sigma (G)\). G is said to be \(\sigma \)-full if G possesses a complete Hall \(\sigma \)-set; \(\sigma \)-nilpotent if G has a complete Hall \(\sigma \)-set \(\mathcal {H}=\{H_1,H_2,\ldots ,H_t\}\) such that \(G=H_1\times H_2\times \cdots \times H_t\); \(\mathfrak {N}_{\sigma }\)-critical (resp. \(\mathfrak {N}\)-critical) if G is not \(\sigma \)-nilpotent (resp. nilpotent) but all proper subgroups of G are \(\sigma \)-nilpotent (resp. nilpotent). An \(\mathfrak {N}\)-critical group is also called a Schmidt group. In this paper, we first prove that every \(\mathfrak {N}_{\sigma }\)-critical group is \(\sigma \)-soluble. This result gives a positive answer to a recent open problem of Skiba. We also prove that \(\mathfrak {N}_{\sigma }\)-critical groups are also Schmidt groups and so the structure of \(\mathfrak {N}_{\sigma }\)-critical groups is obtained.