Let G be a finite group, σ={σi|i∈I}\documentclass[12pt]{minimal}
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\begin{document}$$\sigma =\{\sigma _{i}|i\in I\}$$\end{document} be a partition of the set of all primes P\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {P}$$\end{document} and σ(G)={σi|σi∩π(G)≠∅}\documentclass[12pt]{minimal}
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\begin{document}$$\sigma (G)=\{\sigma _i|\sigma _i\cap \pi (G)\ne \emptyset \}$$\end{document}. G is said to be σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}-primary if |σ(G)|≤1\documentclass[12pt]{minimal}
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\begin{document}$$|\sigma (G)|\le 1$$\end{document}; σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}-soluble if every chief factor of G is σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}-primary. A set H\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {H}$$\end{document} of subgroups of G is said to be a complete Hallσ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}-set of G if every non-identity member of H\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {H}$$\end{document} is a Hall σi\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _i$$\end{document}-subgroup of G for some σi\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _i$$\end{document} and H\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {H}$$\end{document} contains exactly one Hall σi\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _i$$\end{document}-subgroup for every σi∈σ(G)\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _i\in \sigma (G)$$\end{document}. G is said to be σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}-full if G possesses a complete Hall σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}-set; σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}-nilpotent if G has a complete Hall σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}-set H={H1,H2,…,Ht}\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {H}=\{H_1,H_2,\ldots ,H_t\}$$\end{document} such that G=H1×H2×⋯×Ht\documentclass[12pt]{minimal}
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\begin{document}$$G=H_1\times H_2\times \cdots \times H_t$$\end{document}; Nσ\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {N}_{\sigma }$$\end{document}-critical (resp. N\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {N}$$\end{document}-critical) if G is not σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}-nilpotent (resp. nilpotent) but all proper subgroups of G are σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}-nilpotent (resp. nilpotent). An N\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {N}$$\end{document}-critical group is also called a Schmidt group. In this paper, we first prove that every Nσ\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {N}_{\sigma }$$\end{document}-critical group is σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}-soluble. This result gives a positive answer to a recent open problem of Skiba. We also prove that Nσ\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {N}_{\sigma }$$\end{document}-critical groups are also Schmidt groups and so the structure of Nσ\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {N}_{\sigma }$$\end{document}-critical groups is obtained.
