Abstract In this paper, ? is a finite group and ? a partition of the set of all primes ℙ, that is, σ={σi∣i∈I}sigma={sigma_{i}mid iin I} , where P=⋃i∈Iσimathbb{P}=bigcup_{iin I}sigma_{i} and σi∩σj=∅sigma_{i}capsigma_{j}=emptyset for all i≠jineq j . If ? is an integer, we write σ(n)={σi∣σi∩π(n)≠∅}sigma(n)={sigma_{i}midsigma_{i}cappi(n)neqemptyset} and σ(G)=σ(G)sigma(G)=sigma(lvert Grvert) . A group ? is said to be ?-primary if ? is a σisigma_{i} -group for some i=i(G)i=i(G) and ?-soluble if every chief factor of ? is ?-primary. We say that ? is a ?-tower group if either G=1G=1 or ? has a normal series 1=G0展开▼
