- Publication venue
- Publication date
- 28/01/2018
- Field of study
Let G be a finite group and Ο={Οiββ£iβI} some
partition of the set of all primes P, that is, Ο={Οiββ£iβI}, where P=βiβIβΟiβ and Οiββ©Οjβ=β
for all iξ =j. We say that G is Ο-primary
if G is a Οiβ-group for some i. A subgroup A of G is said to
be: Ο-subnormal in G if there is a subgroup chain A=A0ββ€A1ββ€β―β€Anβ=G such that either Aiβ1ββ΄Aiβ
or Aiβ/(Aiβ1β)Aiββ is Ο-primary for all i=1,β¦,n,
modular in G if the following conditions hold: (i) β¨X,Aβ©Zβ©=β¨X,Aβ©β©Z for all Xβ€G,Zβ€G such that Xβ€Z, and (ii) β¨A,Yβ©Zβ©=β¨A,Yβ©β©Z for
all Yβ€G,Zβ€G such that Aβ€Z. In this paper, a subgroup A of
G is called Ο-quasinormal in G if L is modular and
Ο-subnormal in G. We study Ο-quasinormal subgroups of G. In
particular, we prove that if a subgroup H of G is Ο-quasinormal in
G, then for every chief factor H/K of G between HG and HGβ the
semidirect product (H/K)β(G/CGβ(H/K)) is Ο-primary.Comment: 9 page