On ss-quasinormal and weakly s-supplemented subgroups of finite groups
Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. $H$ is called $ss$-quasinormal in $G$ if there is a subgroup $B$ of $G$ such that $G = HB$ and $H$ permutes with every Sylow subgroup of $B$; $H$ is called weakly $s$-supplemented in G if there is a subgroup T of G such that $G = HT$ a...
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| author | Li, C. Li, Yangming Лі, К. Лі, Янмін |
| author_facet | Li, C. Li, Yangming Лі, К. Лі, Янмін |
| author_sort | Li, C. |
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| description | Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. $H$ is called $ss$-quasinormal in $G$ if there is a subgroup $B$ of $G$ such that $G = HB$ and $H$ permutes with every Sylow subgroup of $B$; $H$ is called weakly
$s$-supplemented in G if there is a subgroup T of G such that $G = HT$ and $H \bigcap T \leq H_{sG}$, where $H_{sG}$ is
the subgroup of $H$ generated by all those subgroups of $H$ which are $s$-quasinormal in $G$.
In this paper we investigate the influence of $ss$-quasinormal and weakly $s$-supplemented subgroups on the structure of finite
groups. Some recent results are generalized and unified. |
| first_indexed | 2026-03-24T02:31:10Z |
| format | Article |
| fulltext |
UDC 512.5
C. Li (School Math. Sci., Xuzhou Normal Univ., China),
Y. Li (Guangdong Uni. Education, Guangzhou, China)
ON ss-QUASINORMAL AND WEAKLY
s-SUPPLEMENTED SUBGROUPS OF FINITE GROUPS *
ПРО ss-КВАЗIНОРМАЛЬНI ТА СЛАБКО s-ДОПОВНЮВАНI
ПIДГРУПИ СКIНЧЕННИХ ГРУП
Suppose that G is a finite group and H is a subgroup of G. H is called ss-quasinormal in G if there is a
subgroup B of G such that G = HB and H permutes with every Sylow subgroup of B; H is called weakly
s-supplemented in G if there is a subgroup T of G such that G = HT and H ∩ T ≤ HsG, where HsG is
the subgroup of H generated by all those subgroups of H which are s-quasinormal in G. In this paper we
investigate the influence of ss-quasinormal and weakly s-supplemented subgroups on the structure of finite
groups. Some recent results are generalized and unified.
Нехай G — скiнченна група, а H — пiдгрупа G. Пiдгрупа H називається ss-квазiнормальною в G, якщо
iснує така пiдгрупа B групи G, що G = HB i H є переставною з кожною силовською пiдгрупою
пiдгрупи B; H називається слабко s-доповнюваною в G, якщо iснує така пiдгрупа T групи G, що
G = HT i H ∩ T ≤ HsG, де HsG — пiдгрупа H, що породжена усiма пiдгрупами H, якi є s-
квазiнормальними вG. У данiй роботi дослiджено вплив ss-квазiнормальних та слабко s-доповнюваних
пiдгруп на структуру скiнченних груп. Узагальнено та унiфiковано деякi нещодавнi результати.
1. Introduction. All groups considered in this paper will be finite. We use conventional
notions and notation, as in Huppert [1]. G denotes a group, |G| is the order of G, π(G)
denotes the set of all primes dividing |G| and Gp is a Sylow p-subgroup of G for some
p ∈ π(G). Let F be a class of groups. We call F a formation provided that (i) if G ∈ F
and H E G, then G/H ∈ F , and (ii) if G/M and G/N are in F , then G/(M ∩ N)
is in F for any normal subgroups M, N of G. A formation F is said to be saturated
if G/Φ(G) ∈ F implies that G ∈ F . In this paper, U will denote the class of all
supersolvable groups. Clearly, U is a saturated formation [1, p. 713] (Satz 8.6).
Two subgroups H and K of G are said to be permutable if HK = KH. A sub-
group H of G is said to be s-quasinormal (or s-permutable, π-quasinormal) in G if
H permutes with every Sylow subgroup of G. This concept was introduced by Kegel
in [2]. In 2008, Shirong Li [3] introduced the concept of ss-quasinormal subgroup: A
subgroup H of a group G is said to be an ss-quasinormal subgroup of G if there is
a subgroup B such that G = HB and H permutes with every Sylow subgroup of B.
Groups with certain ss-quasinormal subgroups of prime power order were studied in
[3]. Obviously, s-quasinormal subgroup is ss-quasinormal subgroup. As another gener-
alization of s-quasinormal subgroups, A. N. Skiba [4] introduced the following concept:
A subgroup H of a group G is called weakly s-supplemented in G if there is a subgroup
T of G such that G = HT and H ∩ T ≤ HsG, where HsG is the subgroup of H
generated by all those subgroups of H which are s-quasinormal in G. In fact, this con-
cept is also a generalization of c-normal and c-supplemented subgroups given in [5] and
[6]. A. N. Skiba proposed in [4] two open questions related to weakly s-supplemented
subgroups. In this paper, we prove some theorems which show that in most cases (for
maximal and minimal subgroups) the question 6.4 in [4] has positive solution.
*The project is supported by Natural Sciences Foundation of China (No. 11071229) and the Natural
Sciences Foundation of the Jiangsu Higher Education Institutions (No. 10KJD110004).
c© C. LI, Y. LI, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12 1623
1624 C. Li, Y. Li
There are examples to show that weakly s-supplemented subgroups are not ss-
quasinormal subgroups and in general the converse is also false. The aim of this ar-
ticle is to unify and improve some earlier results using ss-quasinormal and weakly
s-supplemented subgroups. The main result of the paper is Theorem 3.4.
2. Preliminaries.
Lemma 2.1 ([3], Lemma 2.1). LetH be an ss-quasinormal subgroup of a groupG.
(1) If H ≤ L ≤ G, then H is ss-quasinormal in L.
(2) If N �G, then HN/N is ss-quasinormal in G/N.
Lemma 2.2 ([3], Lemma 2.2). Let H be a nilpotent subgroup of G. Then the fol-
lowing statements are equivalent:
(1) H is s-quasinormal in G.
(2) H ≤ F (G) and H is ss-quasinormal in G.
Lemma 2.3 ([3], Lemma 2.5). If a p-subgroup P of G is ss-quasinormal (p a
prime), then P permutes with every Sylow q-subgroup of G with q 6= p.
Lemma 2.4. Let P be a Sylow p-subgroup of a group G, where p is a prime
divisor of |G| with (|G|, p− 1) = 1. If every maximal subgroup of P is ss-quasinormal
in G, then G is p-nilpotent.
Proof. This is a corollary of the proof of [3] (Theorem 1.1) by [25] (Lemma 2.6).
Lemma 2.5 ([4], Lemma 2.10). Let H be a weakly s-supplemented subgroup of a
group G.
(1) If H ≤ L ≤ G, then H is weakly s-supplemented in L.
(2) If N E G and N ≤ H ≤ G, then H/N is weakly s-supplemented in G/N.
(3) If H is a π-subgroup and N is a normal π′-subgroup of G, then HN/N is
weakly s-supplemented in G/N.
Lemma 2.6 ([7], A, 1.2). Let U, V, and W be subgroups of a group G. Then the
following statements are equivalent:
(1) U ∩ VW = (U ∩ V )(U ∩W ).
(2) UV ∩ UW = U(V ∩W ).
Lemma 2.7 ([8], Lemma 2.2.). If P is an s-quasinormal p-subgroup of a group G
for some prime p, then NG(P ) ≥ Op(G).
Lemma 2.8 ([11], Lemma 2.6). Let H be a solvable normal subgroup of a group
G (H 6= 1). If every minimal normal subgroup of G which is contained in H is not
contained in Φ(G), then the Fitting subgroup F (H) of H is the direct product of
minimal normal subgroups of G which are contained in H.
Lemma 2.9 ([4], Lemma 2.16). Let F be a saturated formation containing U , the
class of all supersoluble groups. Suppose that G is a group with a normal subgroup N
such that G/N ∈ F . If N is cyclic, then G ∈ F .
Lemma 2.10 ([20], Lemma 2.3). Let G be a group and N ≤ G.
(1) If N �G, then F ∗(N) ≤ F ∗(G).
(2) IfG 6= 1, then F ∗(G) 6= 1. In fact, F ∗(G)/F (G) = Soc(F (G)CG(F (G))/F (G)).
(3) F ∗(F ∗(G)) = F ∗(G) ≥ F (G). If F ∗(G) is solvable, then F ∗(G) = F (G).
3. Main results.
Theorem 3.1. Let P be a Sylow p-subgroup of a group G, where p is a prime
divisor of |G| with (|G|, p − 1) = 1. If every maximal subgroup of P is either ss-
quasinormal or weakly s-supplemented in G, then G is p-nilpotent.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
ON ss-QUASINORMAL AND WEAKLY s-SUPPLEMENTED SUBGROUPS OF FINITE GROUPS 1625
Proof. Suppose that the theorem is false and let G be a counterexample of minimal
order. We will derive a contradiction in several steps.
(1) G has a unique minimal normal subgroup N and G/N is p-nilpotent. Moreover
Φ(G) = 1.
Let N be a minimal normal subgroup of G. Consider G/N. We will show that G/N
satisfies the hypothesis of the theorem. Let M/N be a maximal subgroup of PN/N.
It is easy to see that M = P1N for some maximal subgroup P1 of P. It follows that
P1 ∩ N = P ∩ N is a Sylow p-subgroup of N. If P1 is ss-quasinormal in G, then
M/N is ss-quasinormal in G/N by Lemma 2.1. If P1 is weakly s-supplemented in
G, then there is a subgroup T of G such that G = P1T and P1 ∩ T ≤ (P1)sG. So
G/N = M/N · TN/N = P1N/N · TN/N. Since
(|N : P1 ∩N |, |N : T ∩N |) = 1,
we have
(P1 ∩N)(T ∩N) = N = N ∩G = N ∩ P1T.
By Lemma 2.6, (P1N) ∩ (TN) = (P1 ∩ T )N. It follows that
(P1N/N) ∩ (TN/N) = (P1N ∩ TN)/N =
= (P1 ∩ T )N/N ≤ (P1)sGN/N ≤ (P1N/N)sG.
HenceM/N is weakly s-supplemented inG/N. Therefore,G/N satisfies the hypothesis
of the theorem. The choice of G yields that G/N is p-nilpotent. Consequently the
uniqueness of N and the fact that Φ(G) = 1 are obvious.
(2) Op′(G) = 1.
If Op′(G) 6= 1, then N ≤ Op′(G) by step (1). Since
G/Op′(G) ∼= (G/N)/(Op′(G)/N)
is p-nilpotent, G is p-nilpotent, a contradiction.
(3) Op(G) = 1.
If Op(G) 6= 1, Step (1) yields N ≤ Op(G) and Φ(Op(G)) ≤ Φ(G) = 1. Therefore,
G has a maximal subgroup M such that G = MN and G/N ∼= M is p-nilpotent. Since
Op(G)∩M is normalized by N and M, Op(G)∩M is normal in G. The uniqueness of
N yields N = Op(G). Clearly, P = N(P ∩M). Furthermore P ∩M < P, thus there
exists a maximal subgroup P1 of P such that P ∩M ≤ P1. Hence P = NP1. By the
hypothesis, P1 is either ss-quasinormal or weakly s-quasinormal in G. If we assume
that P1 is ss-quasinormal in G, then P1Mq is a group for q 6= p by Lemma 2.3. Hence
P1
〈
Mp,Mq|q ∈ π(M), q 6= p
〉
= P1M
is a group. Then P1M = M or G by maximality of M. If P1M = G, then P =
= P ∩ P1M = P1(P ∩ M) = P1, a contradiction. If P1M = M, then P1 ≤ M.
Therefore, P1 ∩N = 1 and N is of prime order. Then the p-nilpotency of G/N implies
the p-nilpotency of G, a contradiction. Therefore we may assume that P1 is weakly
s-supplemented in G. Then there is a subgroup T of G such that G = P1T and P1 ∩
T ≤ (P1)sG. From Lemma 2.7 we have Op(G) ≤ NG((P1)sG). Since (P1)sG is
subnormal in G, we obtain
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
1626 C. Li, Y. Li
P1 ∩ T ≤ (P1)sG ≤ Op(G) = N.
Thus (P1)sG ≤ P1 ∩N and
(P1)sG ≤ ((P1)sG)G = ((P1)sG)O
p(G)P = ((P1)sG)P ≤ (P1 ∩N)P = P1 ∩N ≤ N.
It follows that ((P1)sG)G = 1 or ((P1)sG)G = P1∩N = N. If ((P1)sG)G = P1∩N =
= N, then N ≤ P1 and P = NP1 = P1, a contradiction. If ((P1)sG)G = 1, then
P1∩T = 1 and so |T |p = p. Hence T is p-nilpotent. Let Tp′ be the normal p-complement
of T. Since M is p-nilpotent, we may suppose that M has a normal Hall p′-subgroup
Mp′ and M ≤ NG(Mp′) ≤ G. The maximality of M implies that M = NG(Mp′)
or NG(Mp′) = G. If the latter holds, then Mp′ E G, and Mp′ is actually the normal
p-complement of G, which is contrary to the choice of G. Hence we may assume that
M = NG(Mp′). By applying a deep result of Gross ([9], main theorem) and Feit –
Thompson’s theorem, there exists g ∈ G such that T g
p′ = Mp′ . Hence T g ≤ NG(T g
p′) =
= NG(Mp′) = M. However, Tp′ is normalized by T, so g can be considered as an
element of P1. Thus G = P1T
g = P1M and P = P1(P ∩M) = P1, a contradiction.
(4) G has Hall p′-subgroups and any two subgroups of G are conjugate in G.
If every maximal subgroup of P is ss-quasinormal in G, then G is p-nilpotent by
Lemma 2.4, a contradiction. Thus there is a maximal subgroup P0 of P such that P0 is
weakly s-supplemented in G. Then there exists a subgroup T of G such that G = P0T
and
P0 ∩ T ≤ (P0)sG ≤ Op(G) = 1.
By [10] (Theorem 2.2), G is not simple and G has Hall p′-subgroups. A new application
of the result of Gross ([9], main theorem) and Feit – Thompson’s theorem yields that any
two Hall p′-subgroups of G are conjugate in G.
(5) The final contradiction.
If we suppose that NP < G, then NP satisfies the hypothesis of the theorem.
The choise of G yields that N is p-nilpotent, a contradiction with steps (2) and (3).
Therefore we may assume that G = NP. By step (4), G has Hall p′-subgroups. Then
we may suppose that N has a Hall p′-subgroup Np′ . By Frattini’s argument,
G = NNG(Np′) = (P ∩N)Np′NG(Np′) = (P ∩N)NG(Np′)
and so
P = P ∩G = P ∩ (P ∩N)NG(Np′) = (P ∩N)(P ∩NG(Np′)).
Since NG(Np′) < G, it follows that P ∩NG(Np′) < P. Consider a maximal subgroup
P1 of P such that P ∩ NG(Np′) ≤ P1. Then P = (P ∩ N)P1. By the hypothesis, P1
is either ss-quasinormal or weakly s-supplemented in G. If P1 is ss-quasinormal in G,
then P1NG(Np′) = P1Np′ forms a group by Lemma 2.3. Since |G : P1Np′ | = p and
(|G|, p − 1) = 1, we have P1Np′ � G by [25] (Lemma 2.6). By Frattini’s argument
again, G = P1Np′NG(Np′) = P1NG(Np′) < G, a contradiction. Now assume that P1
is weakly s-supplemented in G. Then there is a subgroup T of G such that G = P1T
and
P1 ∩ T ≤ (P1)sG ≤ Op(G) = 1.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
ON ss-QUASINORMAL AND WEAKLY s-SUPPLEMENTED SUBGROUPS OF FINITE GROUPS 1627
Since |T |p = p, we have that T is p-nilpotent. Let Tp′ be the normal p-complement of
T, then Tp′ is a Hall p′-subgroup of G. By step (4), Tp′ and Np′ are conjugate in G.
Since Tp′ is normalized by T, there exists g ∈ P1 such that T g
p′ = Np′ . Hence
G = (P1T )g = P1T
g = P1NG(T g
p′) = P1NG(Np′)
and
P = P ∩G = P ∩ P1NG(Np′) = P1(P ∩NG(Np′)) ≤ P1,
a contradiction.
Corollary 3.1. Let p be a prime dividing the order of a group G, where p is the
smallest prime divisor of |G| and H a normal subgroup of G such that G/H is p-
nilpotent. If there exists a Sylow p-subgroup P of H such that every maximal subgroup
of P is either ss-quasinormal or weakly s-supplemented in G, then G is p-nilpotent.
Proof. By Lemmas 2.1 and 2.5, every maximal subgroup of P is either ss-quasi-
normal or weakly s-supplemented in H. By Theorem 3.1, H is p-nilpotent. Now, let
Hp′ be the normal p-complement of H. Then Hp′ E G. Assume that Hp′ 6= 1 and
consider G/Hp′ . Applying Lemmas 2.1 and 2.5 it is easy to see that G/Hp′ satisfies
the hypotheses for the normal subgroup H/Hp′ . Therefore by induction G/Hp′ is p-
nilpotent and so G is p-nilpotent. Hence we may assume Hp′ = 1 and therefore H =
= P is a p-group. Since G/H is p-nilpotent, we can consider K/H be the normal p-
complement of G/H. By Schur – Zassenhaus’s theorem, there exists a Hall p′-subgroup
Kp′ of K such that K = HKp′ . A new application of Theorem 3.1 yields that K is
p-nilpotent and so K = H × Kp′ . Hence Kp′ is a normal p-complement of G. This
completes the proof.
Corollary 3.2 ([13], Theorem 3.4). Let G be a group and P a Sylow p-subgroup
of G, where p is the smallest prime dividing |G|. If all maximal subgroups of P are
c-normal in G, then G is p-nilpotent.
Corollary 3.3 ([14], Theorem 3.4). Let G be a group and P a Sylow p-subgroup
of G, where p is the smallest prime dividing |G|. If all maximal subgroups of P are
c-supplemented in G, then G is p-nilpotent.
Corollary 3.4 ([24], Theorem 3.2). Let p be a prime dividing the order of a group
G with (|G|, p − 1) = 1. If there exists a Sylow p-subgroup P of G such that every
maximal subgroup of P is weakly s-permutable in G, then G is p-nilpotent.
Corollary 3.5. Suppose that every maximal subgroup of any Sylow subgroup of a
group G is either ss-quasinormal or weakly s-supplemented in G, then G is a Sylow
tower group of supersolvable type.
Proof. Let p be the smallest prime dividing |G| and P a Sylow p-subgroup of G.
Then every maximal subgroup of P is either ss-quasinormal or weakly s-supplemented
in G. By Theorem 3.1, G is p-nilpotent. Let U be the normal p-complement of G. By
Lemmas 2.1 and 2.5, U satisfies the hypothesis of the corollary. Therefore it follows by
induction that U, and hence G is a Sylow tower group of supersolvable type.
Theorem 3.2. Let F be a saturated formation containing U , the class of all
supersoluble groups. A group G ∈ F if and only if there is a normal subgroup H of G
such that G/H ∈ F and every maximal subgroup of any Sylow subgroup of H is either
ss-quasinormal or weakly s-supplemented in G.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 12
1628 C. Li, Y. Li
Proof. The necessity is obvious. We only need to prove the sufficiency. Suppose
that the assertion is false and let G be a counterexample of minimal order.
(1) By Lemmas 2.1 and 2.5, every maximal subgroup of any Sylow subgroup of H is
either ss-quasinormal or weakly s-supplemented in H. By Corollary 3.4, H is a Sylow
tower group of supersolvable type. Let p be the largest prime divisor of |H| and let P be
a Sylow p-subgroup of H. Then P is normal in G. Let N be a minimal normal subgroup
of G contained in P. We consider G/N. It is easy to see that (G/N,H/N) satisfies the
hypothesis of the theorem. By the minimality of G, we have G/N ∈ F . Since F is a
saturated formation, N is the unique minimal normal subgroup of G contained in P and
N � Φ(G). By Lemma 2.8, it follows that P = F (P ) = N.
(2) Since N E G, we may take a maximal subgroup N1 of N such that N1 E Gp,
where Gp is a Sylow p-subgroup of G. Then N1 is either ss-quasinormal or weakly
s-supplemented in G. If N1 is weakly group s-supplemented in G, then there is a
subgroup T of G such that G = N1T and N1 ∩ T ≤ (N1)sG. Thus G = NT and
N = N ∩ N1T = N1(N ∩ T ). This implies that N ∩ T 6= 1. But since N ∩ T is
normal in G and N is minimal normal in G, N ∩ T = N. It follows that T = G
and so N1 = (N1)sG is s-quasinormal in G. By Lemma 2.7, Op(G) ≤ NG(N1).
Thus N1 E GpO
p(G) = G. It follows that N1 = 1 and so |N | = p. By Lemma 2.9,
G ∈ F , a contradiction. If N1 is ss-quasinormal in G, then N1 is s-quasinormal in G
by Lemma 2.2 and it follows the same contradiction.
Corollary 3.6 ([14], Theorem 4.2). Let F be a saturated formation containing U ,
the class of all supersoluble groups. If there is a normal subgroup H of G such that
G/H ∈ F and every maximal subgroup of any Sylow subgroup of H is c-supplemented
in G, then G ∈ F .
Corollary 3.7 ([3], Theorem 1.5). Let F be a saturated formation containing U ,
the class of all supersoluble groups. If there is a normal subgroup H of G such that
G/H ∈ F and every maximal subgroup of any Sylow subgroup of H is ss-quasinormal
in G, then G ∈ F .
Corollary 3.8 ([11], Theorem 3.3). Let H be a normal subgroup of a group G such
that G/H is supersolvable. If every maximal subgroup of any Sylow subgroup of H is
c-normal in G, then G is supersolvable.
Corollary 3.9 ([6], Theorem 3.3). Let H be a normal subgroup of a group G such
that G/H is supersolvable. If every maximal subgroup of any Sylow subgroup of H is
c-supplemented in G, then G is supersolvable.
Corollary 3.10 ([5], Theorem 4.1). If every maximal subgroup of any Sylow sub-
group of a group G is c-normal in G, then G is supersolvable.
Theorem 3.3. Let F be a saturated formation containing U , the class of all
supersoluble groups. A group G ∈ F if and only if there is a normal subgroup E
of G such that G/E ∈ F and every cyclic subgroup 〈x〉 of any Sylow subgroup of
E with prime order or order 4 (if the Sylow 2-subgroups are non-abelian) is either
ss-quasinormal or weakly s-supplemented in G.
Proof. We need only to prove the sufficiency part since the necessity part is evident.
Suppose that the assertion is false and let G be a counterexample of minimal order. Then
(1) E is solvable.
Let K be any proper subgroup of E. Then |K| < |G| and K/K ∈ U . Let 〈x〉 be
any cyclic subgroup of any Sylow subgroup of K with prime order or order 4 (if the
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ON ss-QUASINORMAL AND WEAKLY s-SUPPLEMENTED SUBGROUPS OF FINITE GROUPS 1629
Sylow 2-subgroups are non-abelian). It is clear that 〈x〉 is also a cyclic subgroup of
a Sylow subgroup of E with prime order or order 4. By the hypothesis, 〈x〉 is either
ss-quasinormal or weakly s-supplemented in G. By Lemmas 2.1 and 2.5, 〈x〉 is either
ss-quasinormal or weakly s-supplemented in K. This shows that the hypothesis still
holds for (U ,K). By the choice of G, K is supersoluble. By [12] (Theorem 3.11.9), E
is solvable.
(2) GF is a p-group, where GF is the F-residual of G. Moreover GF/Φ(GF ) is a
chief factor of G and exp(GF ) = p or exp(GF ) = 4 (if p = 2 and GF is non-abelian).
Since G/E ∈ F , GF ≤ E. Let M be a maximal subgroup of G such that GF *M
(that is, M is an F-abnormal maximal subgroup of G). Then G = ME. We claim that
the hypothesis holds for (F ,M). In fact,
M/M ∩H ∼= ME/E = G/E ∈ F
and by the similar argument as above, we can prove that the hypothesis holds for
(F ,M). By the choice of G, M ∈ F . Thus (2) holds by [12] (Theorem 3.4.2).
(3) 〈x〉 is s-quasinormal in G for any element x ∈ GF .
Let x ∈ GF . Then the order of x is p or 4 by step (2). By the hypothesis, 〈x〉 is
either ss-quasinormal or weakly s-supplemented in G. If 〈x〉 is ss-quasinormal in G,
then 〈x〉 is s-quasinormal in G by Lemma 2.2 since 〈x〉 ≤ GF ≤ Op(G) ≤ F (G). If
〈x〉 is weakly s-supplemented in G, then there is a subgroup T of G such that G = 〈x〉T
and 〈x〉 ∩ T ≤ 〈x〉sG. Hence
GF = GF ∩G = GF ∩ 〈x〉T = 〈x〉(GF ∩ T ).
Since GF/Φ(GF ) is abelian, we have
(GF ∩ T )Φ(GF )/Φ(GF ) E G/Φ(GF ).
Since GF/Φ(GF ) is a chief factor of G, GF∩T ≤ Φ(GF ) or GF = (GF∩T )Φ(GF ) =
= GF ∩T. If GF ∩T ≤ Φ(GF ), then 〈x〉 = GF E G. In this case, 〈x〉 is s-quasinormal
in G. If GF = GF ∩ T, then T = G and so 〈x〉 = 〈x〉sG is s-quasinormal in G.
(4) |GF/Φ(GF )| = p.
Assume that |GF/Φ(GF )| 6= p and let L/Φ(GF ) be any cyclic subgroup of
GF/Φ(GF ). Let x ∈ L\Φ(GF ). Then L = 〈x〉Φ(GF ). Since 〈x〉 is s-quasinormal in G
by step (3), L/Φ(GF ) is s-quasinormal in G/Φ(GF ). It follows from [4] (Lemma 2.11)
that GF/Φ(GF ) has a maximal subgroup which is normal in G/Φ(GF ). But this is
impossible since GF/Φ(GF ) is a chief factor of G. Thus |GF/Φ(GF )| = p.
(5) The final contradiction.
Since
(G/Φ(GF ))/(GF/Φ(GF )) ∼= G/GF ∈ F ,
we have that G/Φ(GF ) ∈ F by Lemma 2.9. But Φ(GF ) ≤ Φ(G) and F is a saturated
formation, therefore G ∈ F , the final contradiction.
Corollary 3.11 ([15], Theorem 4.2). Let F be a saturated formation containing U ,
the class of all supersoluble groups. If every cyclic subgroup of GF of prime order or
order 4 is c-normal in G, then G ∈ F .
Corollary 3.12 ([16], Theorem 4.1). If every cyclic subgroup of GU of prime order
or order 4 is c-supplemented in G, then G is supersolvable.
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1630 C. Li, Y. Li
Corollary 3.13 ([17], Theorem 3.4). If every cyclic subgroup of G of prime order
or order 4 is ss-quasinormal in G, then G is supersolvable.
Corollary 3.14 ([19], Theorem 3.2). Let F be a saturated formation containing U .
A group G ∈ F if and only if there is a normal subgroup E of G such that G/E ∈ F
and every subgroup of E of prime order or order 4 is weakly s-supplemented in G.
Theorem 3.4. Let F be a saturated formation containing U . Suppose that G is a
group with a normal subgroup H such that G/H ∈ F . Then G ∈ F if and only if one
of the following conditions holds:
(1) every maximal subgroup of any Sylow subgroup of F ∗(H) is either ss-quasinormal
or weakly s-supplemented in G.
(2) every cyclic subgroup of any Sylow subgroup of F ∗(H) of prime order or order
4 is either ss-quasinormal or weakly s-supplemented in G.
Proof. We only need to prove the “if” part. If the condition (1) holds, then ev-
ery maximal subgroup of any Sylow subgroup of F ∗(H) is either ss-quasinormal or
weakly s-supplemented in F ∗(H) by Lemmas 2.1 and 2.5. From Theorem 3.2, we
have that F ∗(H) is supersolvable. In particular, F ∗(H) is solvable. By Lemma 2.10,
F ∗(H) = F (H). Since s-quasinormal subgroup is weakly s-supplemented subgroup,
it follows that every maximal subgroup of any Sylow subgroup of F ∗(H) is weakly
s-supplemented in G by Lemma 2.2. Applying [21] (Theorem A), G ∈ F . If the condi-
tion (2) holds, then we have also G ∈ F using similar arguments as above.
Corollary 3.15 ([22], Theorem 3.4). Let F be a saturated formation containing U .
Suppose that G is a group with a normal subgroup H such that G/H ∈ F . If all
maximal subgroups of any Sylow subgroup of F ∗(H) are s-quasinormal in G, then
G ∈ F .
Corollary 3.16 ([20], Theorem 3.1). Let F be a saturated formation containing U .
Suppose that G is a group with a normal subgroup H such that G/H ∈ F . If all
maximal subgroups of any Sylow subgroup of F ∗(H) are c-normal in G, then G ∈ F .
Corollary 3.17 ([23], Theorem 1.1). Let F be a saturated formation containing U .
Suppose that G is a group with a normal subgroup H such that G/H ∈ F . If all
maximal subgroups of any Sylow subgroup of F ∗(H) are c-supplemented in G, then
G ∈ F .
Corollary 3.18 ([17], Theorem 3.3). Let F be a saturated formation containing U .
Suppose that G is a group with a normal subgroup H such that G/H ∈ F . Then
G ∈ F if and only if every maximal subgroup of any Sylow subgroup of F ∗(H) is
ss-quasinormal in G.
Corollary 3.19 ([18], Theorem 3.3). Let F be a saturated formation containing U .
Suppose that G is a group with a normal subgroup H such that G/H ∈ F . If every
cyclic subgroup of any Sylow subgroup of F ∗(H) of prime order or order 4 is s-
quasinormal in G, then G ∈ F .
Corollary 3.20 ([20], Theorem 3.2). Let F be a saturated formation containing U .
Suppose that G is a group with a normal subgroup H such that G/H ∈ F . If every
cyclic subgroup of any Sylow subgroup of F ∗(H) of prime order or order 4 is c-normal
in G, then G ∈ F .
Corollary 3.21 ([23], Theorem 1.2). Let F be a saturated formation containing U .
Suppose that G is a group with a normal subgroup H such that G/H ∈ F . If every
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ON ss-QUASINORMAL AND WEAKLY s-SUPPLEMENTED SUBGROUPS OF FINITE GROUPS 1631
cyclic subgroup of any Sylow subgroup of F ∗(H) of prime order or order 4 is c-
supplemented in G, then G ∈ F .
Corollary 3.22 ([17], Theorem 3.7). Let F be a saturated formation containing U .
Suppose that G is a group with a normal subgroup H such that G/H ∈ F . Then G ∈ F
if and only if every cyclic subgroup of any Sylow subgroup of F ∗(H) of prime order or
order 4 is ss-quasinormal in G.
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Received 13.02.11
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| id | umjimathkievua-article-2830 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:31:10Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/fe/2b85560631e51067c7f52c6b0ff754fe.pdf |
| spelling | umjimathkievua-article-28302020-03-18T19:37:39Z On ss-quasinormal and weakly s-supplemented subgroups of finite groups Про ss-квазiнормальнi та слабко s-доповнюванi пiдгрупи скiнченних груп Li, C. Li, Yangming Лі, К. Лі, Янмін Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. $H$ is called $ss$-quasinormal in $G$ if there is a subgroup $B$ of $G$ such that $G = HB$ and $H$ permutes with every Sylow subgroup of $B$; $H$ is called weakly $s$-supplemented in G if there is a subgroup T of G such that $G = HT$ and $H \bigcap T \leq H_{sG}$, where $H_{sG}$ is the subgroup of $H$ generated by all those subgroups of $H$ which are $s$-quasinormal in $G$. In this paper we investigate the influence of $ss$-quasinormal and weakly $s$-supplemented subgroups on the structure of finite groups. Some recent results are generalized and unified. Нехай $G$ — скiнченна група, а $H$ — пiдгрупа $G$. Пiдгрупа $H$ називається $ss$-квазiнормальною в $G$, якщо iснує така пiдгрупа B групи $G$, що $G = HB$ i $H$ є переставною з кожною силовською пiдгрупою пiдгрупи $B$; $H$ називається слабко $s$-доповнюваною в $G$, якщо iснує така пiдгрупа $T$ групи $G$, що $G = HT$ i $H \bigcap T \leq H_{sG}$, де $H_{sG}$ — пiдгрупа $H$, що породжена усiма пiдгрупами $H$, якi є $s$-квазiнормальними в $G$. У данiй роботi дослiджено вплив $ss$-квазiнормальних та слабко $s$-доповнюваних пiдгруп на структуру скiнченних груп. Узагальнено та унiфiковано деякi нещодавнi результати. Institute of Mathematics, NAS of Ukraine 2011-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2830 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 12 (2011); 1623-1631 Український математичний журнал; Том 63 № 12 (2011); 1623-1631 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2830/2413 https://umj.imath.kiev.ua/index.php/umj/article/view/2830/2414 Copyright (c) 2011 Li C.; Li Yangming |
| spellingShingle | Li, C. Li, Yangming Лі, К. Лі, Янмін On ss-quasinormal and weakly s-supplemented subgroups of finite groups |
| title | On ss-quasinormal and weakly s-supplemented subgroups of finite groups |
| title_alt | Про ss-квазiнормальнi та слабко s-доповнюванi пiдгрупи скiнченних груп |
| title_full | On ss-quasinormal and weakly s-supplemented subgroups of finite groups |
| title_fullStr | On ss-quasinormal and weakly s-supplemented subgroups of finite groups |
| title_full_unstemmed | On ss-quasinormal and weakly s-supplemented subgroups of finite groups |
| title_short | On ss-quasinormal and weakly s-supplemented subgroups of finite groups |
| title_sort | on ss-quasinormal and weakly s-supplemented subgroups of finite groups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2830 |
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