| All groups considered in this thesis are finite.We call a non-nilpotent group in which all proper subgroups are nilpotent Schmidt group.Let σ={σ*|i ∈I} be a partition of the set of all primes P,i.e.,P=∪i∈Iσi for all i≠j,σi∩σj=(?).We say that a subgroup H of G is σ-subnormal in G if there is a chain H=H0≤H1≤…≤Hn=G,such that either Hi-1 (?) Hi or Hi/CoreHi(Hi-1)is σ-primary for i=1,2,…,n.For every partition σ,if G is not a σ-nilpotent group,but all proper subgroups of G are σ-nilpotent,we call group G Nσ-critical.In this paper,we apply the concepts and properties of the σ-subnormal subgroups proposed by Skiba,discuss the cases of arbitrary and binary partition σ with minimal order counterexamples,and obtain some important properties and conclusions.Based on these results,we weaken the nilpotence of the Schmidt subgroups and further study structure of the finite groups that minimal σ-non-nilpotent groups that is Nσcritical subgroups are all σ-subnormal. |
