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Let  G  be a finite group andbe a partition of the set of all primes, that is,andfor all. The natural numbers  n  and  m  are called  σ -coprime if. The group  G  is said to be:  σ -primary if  G  is a  σ i-group for some;  σ -soluble if either  G  = 1 or every chief factor of  G  is  σ -primary. A subgroup  H  of  G  is called  σ -subnormal in  G  if there is a subgroup chainsuch that eitheris normal in  H ioris  σ -primary for all. In this paper, we show that  G  is  σ -soluble provided  G  satisfies the following conditions: (1), where  A1 ,  A2 ,  A3  are all  σ -soluble; (2) the three indicesare pairwise  σ -coprime. And we prove that if  G  is  σ -soluble and  A  is a subgroup of  G , then  A  is  σ -subnormal in  G  if and only ifdividesfor every Hall  σ i-subgroup  B  of  G  and all. We also state and prove a  σ -nilpotency criterion for  G  and a characterization of the  σ -Fitting subgroup of  G , which are related to this observation.

Some notes on σ -soluble groups and σ -subnormality