ISSN 0001-4346, Mathematical Notes, 2018, Vol. 103, No. 2, pp. 313–315. © Pleiades Publishing, Ltd., 2018.
Notes on the Paper “On SS-Quasinormal
and S-Quasinormally Embedded Subgroups
of Finite Groups” of Shen et al.
, and Yuemei Mao
Jiangsu Normal University, Xuzhou, China
Zhejiang Sci-Tech University, Hangzhou, China
University of Datong of Shanxi, Datong, China
Received April 12, 2015; in ﬁnal form, August 16, 2016
Abstract—We correct an error in the paper of Z. Shen, S. Li, and J. Zhang published in . In
addition, we give an answer to a question posed by the authors.
Keywords: S-quasinormally embedded, S-quasinormal, Sylow subgroups.
All groups mentioned in this paper are ﬁnite. Kegel  called a subgroup H of GS-quasinormal
in G if H permutes with every Sylow subgroup of G. We say, following Ballester-Bolinches and
Pedraza-Aquilera , that a subgroup H of G is S-quasinormally embedded in G if a Sylow p-subgroup
of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G for each prime p dividing |H|.In
2008, S. Li etc.  gave the following deﬁnition: A subgroup H of G is said to be an SS-quasinormal
subgroup (supplement-Sylow-quasinormal subgroup)ofG if there is a supplement B of H in G such
that H permutes with every Sylow subgroup of B.
In , the authors proved a number of results; see [4, Theorems 1.1, 1.2, 1.3, 1.4, 1.5, 1.6]. In order
to prove [4, Theorem 1.3], the authors ﬁrst proved [4, Theorems 3.1, 3.2, 3.3]). In the ﬁnal proof of [4,
Theorem 1.3], the authors say “Applying Theorem 3.2, we conclude that G belongs to F.”Since [4,
Theorem 3.2] has the condition that “E is solvable”, we cannot use [4, Theorems 3.2] to prove [4,
Theorem 1.3]. Similarly, the proof of [4, Theorem 1.6] is not true. In Sec. 2, we will correct the proof
of [4, Theorem 1.3] and our proof is simpler.
Since [4, Theorems 1.4, 1.5, 1.6]) have the assumption that |G| is odd, the authors posed the following
question: If the condition that G is a group of odd order is omitted, are Theorems 1.4, 1.5, 1.6 true?
(see [4, P269, Remark]). In Sec. 3, we will give an a
ﬃrmative answer to this question.
2. CORRECTED PROOF OF [4, THEOREM 1.3]
Let Q be a Sylow q-subgroup of F
(E),whereq is the smallest prime divisor of |F
cyclic, then F
(E) is, obviously, q-nilpotent. We assume that Q is noncyclic. In view of [2, Lemma 1],
all subgroups H of Q of order |H| = |D| and 2|D| (if Q is a non-Abelian 2-group and |P : D| > 2)not
having a q-nilpotent supplement in F
(E) are SS-quasinormal in F
(E). By [4, Theorem 1.2], F
is q-nilpotent. It follows that F
(E) is soluble from the well-known Feit–Thompson Theorem and so
(E)=F (E). In view of [3, Theorem 2.2], all subgroups H of noncyclic Sylow subgroup P of F
of order |H| = |D| and 2|D| (if P is a non-Abelian 2-group and |P : D| > 2) not having a supersolvable
supplement in G are S-quasinormal in G. If some subgroup of F
(E) has a supersolvable supplement
(E) ∈F. By [5, Theorem 1.4], G ∈F. Hence we may assume all subgroups H of
noncyclic Sylow subgroup P of F
(E) of order |H| = |D| and 2|D| (if P is a non-Abelian 2-group and
|P : D| > 2)areS-quasinormal in G. Applying [5, Theorem 1.3], we obtain G ∈F.
The article was submitted by the authors for the English version of the journal.