On S-quasinormally embedded subgroups of finite groups
Let G be a finite group. A subgroup A is called:1) S-quasinormal in G if A is permutable with all Sylow subgroups in G 2) S-quasinormally embedded in G if every Sylow subgroup of A is a Sylow subgroup of some S-quasinormal subgroup of G. Let BseG be the subgroup generated by all the subgroups of B w...
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| citation_txt | On S-quasinormally embedded subgroups of finite groups / Kh.A. Al-Sharo, O. Shemetkova, Xiaolan Yi // Algebra and Discrete Mathematics. — 2012. — Vol. 13, № 1. — С. 18–25. — Бібліогр.: 20 назв. — англ. |
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| description | Let G be a finite group. A subgroup A is called:1) S-quasinormal in G if A is permutable with all Sylow subgroups in G 2) S-quasinormally embedded in G if every Sylow subgroup of A is a Sylow subgroup of some S-quasinormal subgroup of G. Let BseG be the subgroup generated by all the subgroups of B which are S-quasinormally embedded in G. A subgroup B is called SE-supplemented in G if there exists a subgroup T such that G = BT and B ∩ T ≤ BseG. The main result of the paper is the following.
|
| first_indexed | 2025-12-07T15:59:30Z |
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h.Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 13 (2012). Number 1. pp. 18 – 25
c© Journal “Algebra and Discrete Mathematics”
On S-quasinormally embedded subgroups
of finite groups
Kh. A. Al-Sharo, Olga Shemetkova and Xiaolan Yi1
Communicated by V. V. Kirichenko
Abstract. Let G be a finite group. A subgroup A is called:
1) S-quasinormal in G if A is permutable with all Sylow subgroups
in G 2) S-quasinormally embedded in G if every Sylow subgroup of
A is a Sylow subgroup of some S-quasinormal subgroup of G. Let
BseG be the subgroup generated by all the subgroups of B which
are S-quasinormally embedded in G. A subgroup B is called SE-
supplemented in G if there exists a subgroup T such that G = BT
and B ∩ T ≤ BseG. The main result of the paper is the following.
Theorem. Let H be a normal subgroup in G, and p a prime
divisor of |H| such that (p−1, |H|) = 1. Let P be a Sylow p-subgroup
in H. Assume that all maximal subgroups in P are SE-supplemented
in G. Then H is p-nilpotent and all its G-chief p-factors are cyclic.
1. Introduction
All groups considered in this paper will be finite. A subgroup A of a
group G is said to be S-quasinormal in G if it permutes with every Sylow
subgroup of G. This concept was introduced by Kegel in [1] and has been
studied in [2]–[15]. In 1998, Ballester-Bolinches and Pedraza-Aguilera [3]
introduced the following definition: A subgroup B of a group G is said
to be S-quasinormally embedded in G if for each prime p dividing the
1Research of the third author (corresponding author) was supported by NNSF of
China (grant no. 11101369).
2010 Mathematics Subject Classification: 20D10, 20D20, 20D25.
Key words and phrases: Finite group, p-nilpotent, S-quasinormal subgroup.
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Kh. A. Al-Sharo, O. Shemetkova, Xiaolan Yi 19
order of B, a Sylow p-subgroup of B is also a Sylow p-subgroup of some
S-quasinormal subgroup of G. In 2007, Al-Sharo and Shemetkova proved
the following.
Theorem 1. Let H be a normal subgroup of a group G, and let p be the
smallest prime dividing |H|. Let P be a Sylow p-subgroup of H. Assume
that every maximal subgroup of P is S-quasinormally embedded in G. Then
H is p-nilpotent and its non-Frattini G-chief p-factors are cyclic (see [10,
Theorem 1.2]).
In 2007, Skiba introduced [11] the concept of S-core as follows.
Definition 1. Let B be a subgroup of a group G. Let BsG be the subgroup
generated by all the subgroups of B which are S-quasinormal in G. The
subgroup BsG is called the S-core of H in G.
A subgroup B of G is called S-supplemented in G if there exists a
subgroup T such that G = BT and B ∩ T ≤ BsG.
By using the concept of S-supplemented subgroup, Skiba proved the
following important result.
Theorem 2. Let E be a normal subgroup of a group G. Suppose that for
every non-cyclic Sylow subgroup P of E, all maximal subgroups of P are
S-supplemented in G. Then each G-chief factor of E is cyclic (see [13,
Theorem A]).
Recently, based on the concept of S-quasinormally embedded subgroup,
Skiba introduced [14] the following.
Definition 2. Let B be a subgroup of a group G. Let BseG be the
subgroup generated by all the subgroups of B which are S-quasinormally
embedded in G. The subgroup BseG is called the SE-core of B in G.
A subgroup B of G is called SE-supplemented in G if there exists a
subgroup T such that G = BT and B ∩ T ≤ BseG.
In the present paper, by using the concept of SE-supplemented sub-
group, we will prove the following improvement of Theorem 1.
Theorem 3. Let H be a normal subgroup in G, and p a prime divisor of
|H| such that (p− 1, |H|) = 1. Let P be a Sylow p-subgroup in H. Assume
that all maximal subgroups in P are SE-supplemented in G. Then H is
p-nilpotent and all its G-chief p-factors are cyclic.
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20 On S-quasinormally embedded. . .
Corollary 1. Let H be a normal subgroup in G, and p a prime divisor of
|H| such that (p− 1, |H|) = 1. Let P be a Sylow p-subgroup in H. Assume
that all maximal subgroups in P are S-supplemented in G. Then H is
p-nilpotent and all its G-chief p-factors are cyclic.
Theorem 2 can be easily deduced from Corollary 1 though we should
notice that Theorem 2 is used in the proof of Theorem 3. The next corollary
is a strengthened version of Theorem 1.
Corollary 2. Let H be a normal subgroup in G, and p a prime divisor
of |H| such that (p − 1, |H|) = 1. Let P be a Sylow p-subgroup in H.
Assume that all maximal subgroups in P are S-quasinormally embedded
in G. Then H is p-nilpotent and all its G-chief p-factors are cyclic.
2. Preliminaries
We use standard notations (see [16]). A subgroup T is called a sup-
plement to a subgroup B in a group G if G = BT . We denote by HG
the core of H in G, the largest normal subgroup of G contained in H. A
group (a subgroup) S is called a Schmidt group (a Schmidt subgroup) if
every proper subgroup of S is nilpotent. We denote by π(G) the set of
all prime divisors of |G|. A group G is called p-supersoluble if every chief
p-factor of G is cyclic.
Lemma 1. Let G be a group and H ≤ K ≤ G.
(1) If H is S-quasinormal in G, then H is S-quasinormal in K.
(2) If H EG, then K/H is S-quasinormal in G/H if and only if K
is S-quasinormal in G.
(3) If H is S-quasinormal in G, then H is subnormal in G.
(4) If A and B are S-quasinormal in G, then A ∩B and 〈A,B〉 are
S-quasinormal in G (see [1]).
Lemma 2. Let A,B be some subgroups in G.
(1) If A is S-quasinormal in G, then A ∩B is S-quasinormal in B.
(2) If If A is S-quasinormal in G, then A/AG is nilpotent (see [2]).
Lemma 3. Suppose that a subgroup U is S-quasinormally embedded in a
group G. Let H ≤ G, and K be a normal subgroup of G. Then:
(a) If U ≤ H, then U is S-quasinormally embedded in H.
(b) UK is S-quasinormally embedded in G, and UK/K is S -quasi-
normally embedded in G/K (see [3]).
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Kh. A. Al-Sharo, O. Shemetkova, Xiaolan Yi 21
Lemma 4. Let H be an SE-supplemented subgroup of G, and N a normal
subgroup in G.
(1) If H ≤ K ≤ G, then H is SE-supplemented in K.
(2) If N ≤ H, then H/N is SE-supplemented in G/N .
(3) If (|N |, |H|) = 1, then HN/N is SE-supplemented in G/N (see
[14, Lemma 2.8]).
The following result is well known.
Lemma 5. Let p be a prime divisor of G such that (p− 1, |G|) = 1.
(1) If M ≤ G and |G : M | = p, then M is normal in G.
(2) If a Sylow p-subgroup of G is cyclic, then G is p-nilpotent.
(3) If G is p-supersoluble, then G is p-nilpotent.
Lemma 6. If a p-subgroup H is S-quasinormal in G, then H ≤ Op(G)
and Op(G) ≤ NG(H) (see [15]).
Lemma 7. If G is a Schmidt group, then:
(1) G is a p-closed {p, q}-group for some primes p, q;
(2) if P is a Sylow p-subgroup of G, then P/Φ(P ) is a chief factor of
G and |P/Φ(P )| = pn ≡ 1 (mod q) where n is the order of p modulo q
(see [17, Theorem 26.1]) and [16, Theorem VII.6.18]).
Lemma 8. Let R E G. Assume that R/Op′(G) is not contained in the
hypercentre of G/Op′(G). Then G has a p-closed Schmidt subgroup S such
that a Sylow p-subgroup Sp 6= 1 of S is contained in R (see [18, Lemma 3]).
Lemma 9. Let p be a prime divisor of G such that (p− 1, |G|) = 1. Let
Gp be a Sylow p-subgroup of G, K EG, P = Gp ∩K. If G/K is a p-group
and every maximal subgroup of Gp either contains P or has a p-nilpotent
supplement in G, then K is p-nilpotent.
Proof. Assume that K is not p-nilpotent. Then by [20, Theorem IV.4.7] we
have P 6≤ Φ(Gp). Let M1 be a maximal subgroup in Gp not containing P .
It follows that there exists a p-nilpotent subgroup T1 such that G = M1T1.
Clearly, Gp = M1(Gp ∩ T1), and we can assume that T1 = NG(H1) where
H1 is a Hall p′-subgroup of K. We see that by [19] every two Hall p′-
subgroup of K are conjugate in K (by assumption, either p = 2 or |G|
is odd). By Frattini argument, G = KT1 = PT1, hence Gp = P (Gp ∩ T1)
and Gp ∩ T1 6≤ P . Let M2 be a maximal subgroup in Gp containing
Gp ∩ T1. Then G = M2T2 where T2 is the normalizer in G of some Hall
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22 On S-quasinormally embedded. . .
p′-subgroup H2 of K. Since Hx
1
= H2, T
x
1
= T2 for some x ∈ G, it follows
that G = M2T2 = M2T
x
1
= M1T1 = M2T1. Therefore
Gp = M1(Gp ∩ T1) = M2(Gp ∩ T1) = M2,
a contradiction.
3. Proof of Theorem 3
Suppose that the theorem is not true and choose a counterexample
(G,H) for which |G|+ |H| is minimal. We will prove several propositions
and will get a contradiction. It follows from Lemma 5 that P is non-cyclic.
(1) Op′(H) = 1.
Assume that Op′(H) 6= 1. Applying Lemma 4 we see that the theo-
rem is true for (G/Op′(H), H/Op′(H)), and then it is true for (G,H), a
contradiction.
(2) H = G.
Assume that H 6= G. By Lemma 4 the theorem is true for the pair
(H,H). Hence H is p-nilpotent. It follows by (1) that H is a p-group. By
Theorem 2 every G-chief factor of H is cyclic, a contradiction.
From (1) and (2) we get the following.
(3) Op′(G) = 1.
(4) |P | > p2.
Assume that |P | = p2. Applying Lemma 5 and Lemma 8 we see that
P is contained in a p-closed Schmidt subgroup S of order p2qb where q is
a prime and p2 ≡ 1 (mod q). Clearly, a Sylow q-subgroup of S is maximal
in S. By Lemma 4 all subgroups of order p in P are SE-supplemented
in S. Applying Lemmas 1 and 3 we see that all subgroups of order p in
P are S-quasinormal in S. Therefore S has a subgroup of order pqb, a
contradiction.
(5) P is non-normal in G.
Assume that P is normal in G. Since the theorem is true for (G,P ),
G is p-supersoluble and so p-nilpotent by Lemma 5, a contradiction.
The following two propositions follow from Lemma 4 and the minimal-
ity of the counterexample G.
(6) If N is minimal normal subgroup in G contained in P , then G/N
is p-supersoluble.
(7) If P ≤ M < G, then M is p-nilpotent.
(8) G is p-soluble.
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Kh. A. Al-Sharo, O. Shemetkova, Xiaolan Yi 23
Assume that G is not p-soluble. By Lemma 6 the unit subgroup 1 is
the only S-quasinormal subgroup contained in P . In particular, PG = 1.
Since (p− 1, |G|) = 1, we have p = 2. By (7) there is a unique minimal
normal subgroup K in G, and PK = G.
Let M be a maximal subgroup in P such that M 6≥ P ∩ K. Since
M is SE-supplemented in G, there is a subgroup T such that G = MT
and M ∩ T ≤ MseG. If MseG = 1, we have |T |2 = 2, and therefore T
is 2-nilpotent. Assume that MseG 6= 1. Then there exists a non-identity
subgroup L in M such that L is S-quasinormally embedded in G. Therefore
L is a Sylow p-subgroup of some S-qusinormal subgroup D. If DG = 1,
it follows that D is nilpotent by Lemma 2. Then by Lemma 6 we have
F (G) 6= 1, which contradicts (3) and PG = 1. Therefore K ≤ DG 6= 1
and L ≥ P ∩ K. So we proved that every maximal subgroup in P not
containing P ∩K has a 2-nilpotent supplement. By Lemma 9 we have
that K is 2-nilpotent, and (8) is proved.
The final contradiction.
From (1–8) it follows that G has a unique minimal normal subgroup
K, and the following properties are valid: 1) K is a p-group and K 6= P ;
2) G/K is p-nilpotent; 3) K = CG(K) = F (G).
Let M be a maximal subgroup in P such that M 6≥ K. Since M
is SE-supplemented in G, there is a subgroup T such that G = MT
and M ∩ T ≤ MseG. If MseG = 1, we have |T |p = p, and therefore T
is p-nilpotent. Assume that MseG 6= 1. Then there exists a non-identity
subgroup L in M such that L is S-quasinormally embedded in G. Therefore
L is a Sylow p-subgroup of some S-qusinormal subgroup D. If DG 6= 1,
then K ≤ DG and K ≤ L ≤ M , a contradiction. Let DG = 1. Then by
Lemma 2 we have that D is nilpotent, and so L = D is an S-qusinormal p-
subgroup. By Lemma 6 we have that Op(G) ≤ NG(L). So, from L ≤ MEP
and G = POp(G) it follows that
K ≤ 〈Lx | x ∈ G〉 = 〈Lx | x ∈ P 〉 ≤ M,
a contradiction. We proved that every maximal subgroup in P not con-
taining K has a p-nilpotent supplement in G. But then by Lemma 9 we
have that KQ is p-nilpotent.
The proof of Theorem 3 is completed.
References
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24 On S-quasinormally embedded. . .
[2] W. E. Deskins, On quasinormal subgroups of finite groups, Math. Z. 82 (1963),
125–132.
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Kh. A. Al-Sharo, O. Shemetkova, Xiaolan Yi 25
Contact information
Kh. A. Al-Sharo Al al-Bayt University, St. Al-Zohoor 5–3, Mafraq
25113, Jordan
E-Mail: sharo_kh@yahoo.com
O. L. Shemetkova Russian Economic University named after
G. V. Plekhanov, Stremyanny Per., 36, 117997
Moscow, Russia
E-Mail: ol-shem@mail.ru
Xiaolan Yi Zhejiang Sci-Tech University, Hangzhou 310018,
P. R. China
E-Mail: yixiaolan2005@126.com
Received by the editors: 31.01.2012
and in final form 31.01.2012.
Kh. A. Al-Sharo, O. Shemetkova, Xiaolan Yi
|
| id | nasplib_isofts_kiev_ua-123456789-152183 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T15:59:30Z |
| publishDate | 2012 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Al-Sharo, Kh.A. Shemetkova, O. Xiaolan Yi 2019-06-08T09:40:28Z 2019-06-08T09:40:28Z 2012 On S-quasinormally embedded subgroups of finite groups / Kh.A. Al-Sharo, O. Shemetkova, Xiaolan Yi // Algebra and Discrete Mathematics. — 2012. — Vol. 13, № 1. — С. 18–25. — Бібліогр.: 20 назв. — англ. 1726-3255 2010 Mathematics Subject Classification:20D10, 20D20, 20D25. https://nasplib.isofts.kiev.ua/handle/123456789/152183 Let G be a finite group. A subgroup A is called:1) S-quasinormal in G if A is permutable with all Sylow subgroups in G 2) S-quasinormally embedded in G if every Sylow subgroup of A is a Sylow subgroup of some S-quasinormal subgroup of G. Let BseG be the subgroup generated by all the subgroups of B which are S-quasinormally embedded in G. A subgroup B is called SE-supplemented in G if there exists a subgroup T such that G = BT and B ∩ T ≤ BseG. The main result of the paper is the following. Research of the third author (corresponding author) was supported by NNSF of China (grant no. 11101369). en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On S-quasinormally embedded subgroups of finite groups Article published earlier |
| spellingShingle | On S-quasinormally embedded subgroups of finite groups Al-Sharo, Kh.A. Shemetkova, O. Xiaolan Yi |
| title | On S-quasinormally embedded subgroups of finite groups |
| title_full | On S-quasinormally embedded subgroups of finite groups |
| title_fullStr | On S-quasinormally embedded subgroups of finite groups |
| title_full_unstemmed | On S-quasinormally embedded subgroups of finite groups |
| title_short | On S-quasinormally embedded subgroups of finite groups |
| title_sort | on s-quasinormally embedded subgroups of finite groups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/152183 |
| work_keys_str_mv | AT alsharokha onsquasinormallyembeddedsubgroupsoffinitegroups AT shemetkovao onsquasinormallyembeddedsubgroupsoffinitegroups AT xiaolanyi onsquasinormallyembeddedsubgroupsoffinitegroups |