text: abstract
  let g be a finite groupsand σ = {σi |i ∈ i} some partition of the set of all primes p, that is,
  σ = {σi |i ∈ i}, where p = i∈i σi and σi ∩ σj = ∅ for all i 6= j. we say that g is σ-primary if g
  is a σi -group for some i. a subgroup a of g is said to be: σ-subnormal in g if there is a subgroup
  chain a = a0 ≤ a1 ≤ · · · ≤ an = g such that either ai−1 e ai or ai /(ai−1 )ai is σ-primary for
  all i = 1, . . . , n; modular in g if the following conditions hold: (i) hx, a ∩ zi = hx, ai ∩ z for all
  x ≤ g, z ≤ g such that x ≤ z, and (ii) ha, y ∩ zi = ha, y i ∩ z for all y ≤ g, z ≤ g such
  that a ≤ z.
  in this paper, a subgroup a of g is called σ-quasinormal in g if l is modular and σ-subnormal
  in g.
  we study σ-quasinormal subgroups of g. in particular, we prove that if a subgroup h of g
  is σ-quasinormal in g, then for every chief factor h/k of g between h g and hg the semidirect
  product (h/k) ⋊ (g/cg (h/k)) is σ-primary.