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

Abstract

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.