Weakly SS-Quasinormal Minimal Subgroups and the Nilpotency of a Finite Group
A subgroup H is said to be an s-permutable subgroup of a finite group G provided that the equality HP =PH holds for every Sylow subgroup P of G. Moreover, H is called SS-quasinormal in G if there exists a supplement B of H to G such that H permutes with every Sylow subgroup of B. We show that H is w...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508055520673792 |
|---|---|
| author | Zhang, Xirong Zhao, Tao Чжан, Хіронг Чжао, Тао |
| author_facet | Zhang, Xirong Zhao, Tao Чжан, Хіронг Чжао, Тао |
| author_sort | Zhang, Xirong |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T10:24:43Z |
| description | A subgroup H is said to be an s-permutable subgroup of a finite group G provided that the equality HP =PH holds for every Sylow subgroup P of G. Moreover, H is called SS-quasinormal in G if there exists a supplement B of H to G such that H permutes with every Sylow subgroup of B. We show that H is weakly SS-quasinormal in G if there exists a normal subgroup T of G such that HT is s-permutable and H \ T is SS-quasinormal in G. We study the influence of some weakly SS-quasinormal minimal subgroups on the nilpotency of a finite group G. Numerous results known from the literature are unified and generalized. |
| first_indexed | 2026-03-24T02:19:07Z |
| format | Article |
| fulltext |
UDC 512.5
T. Zhao (School Sci., Shandong Univ. Technology, China),
X. Zhang (School Math. Sci., Huaiyin Normal Univ., Jiangsu, China)
WEAKLY SS-QUASINORMAL MINIMAL SUBGROUPS
AND THE NILPOTENCY OF A FINITE GROUP*
СЛАБКО SS-КВАЗIНОРМАЛЬНI МIНIМАЛЬНI ПIДГРУПИ
ТА НIЛЬПОТЕНТНIСТЬ СКIНЧЕННОЇ ГРУПИ
A subgroup H is said to be an s-permutable subgroup of a finite group G provided that HP = PH holds for every Sylow
subgroup P of G, and H is said to be SS-quasinormal in G if there is a supplement B of H to G such that H permutes
with every Sylow subgroup of B. We show that H is weakly SS-quasinormal in G if there exists a normal subgroup T
of G such that HT is s-permutable and H ∩ T is SS-quasinormal in G. We investigate the influence of some weakly
SS-quasinormal minimal subgroups on the nilpotency of a finite group G. Numerous results known from the literature are
unified and generalized.
Пiдгрупа H називається s-переставною пiдгрупою скiнченної групи G за умови, що HP = PH виконується
для кожної силовської пiдгрупи P групи G; H називається SS-квазiнормальною в G, якщо iснує доповнення
B пiдгрупи H до G таке, що H можна переставити з кожною силовською пiдгрупою B. Показано, що H є
слабко SS-квазiнормальною в G, якщо iснує нормальна пiдгрупа T групи G така, що HT є s-переставною, а
H ∩ T є SS-квазiнормальною в G. Дослiджено вплив деяких слабко SS-квазiнормальних мiнiмальних пiдгруп на
нiльпотентнiсть скiнченної групиG. Велику кiлькiсть вiдомих з лiтератури результатiв упорядковано та узагальнено.
1. Introduction. All groups considered in this paper will be finite and we use conventional notions
and notation, as in D. Gorenstein [7]. We use F to denote a formation, N and Np denote the classes
of all nilpotent groups and p-nilpotent groups, respectively. GF is the F-residual of G, that is,
GF = ∩{N �G
∣∣G/N ∈ F}. A normal subgroup N is said to be F-hypercentral in G, provided that
N has a chain of subgroups 1 = N0 �N1 � . . .�Nr = N such that each Ni+1/Ni is an F-central
chief factor of G. The product of all F-hypercentral subgroups of G is again an F-hypercentral
subgroup of G, it is denoted by ZF (G) and called the F-hypercenter of G. For the formation N , we
use the notation ZN (G) = Z∞(G), which is the hypercenter of G.
In the study of group theory, from the generalized normalities of some primary subgroups to
investigate the structures of a finite group is a common method. Recently, many new generalized
normal subgroups were introduced successively. Following Kegel [12], a subgroup H is said to be
s-permutable in G, if H is permutable with every Sylow subgroup P of G. As a development, in
[13] the authors introduced that: a subgroup H is called an SS-quasinormal subgroup of G if there
is a supplement B of H to G such that H permutes with every Sylow subgroup of B. Recently, in
[8] Guo et al. introduced that: a subgroup H is said to be S-embedded in G if there exists a normal
subgroup N such that HN is s-permutable in G and H ∩ N ≤ HsG, where HsG is the largest s-
permutable subgroup of G contained H . This concept integrated both the s-permutability and another
related concept called c-normal subgroup, introduced by Wang in [18] and investigated extensively
by many scholars. By assuming that some primary subgroups of G satisfying the s-permutability,
SS-quasinormality or S-embedded properties, many interesting results have been derived (see, for
example, [1, 8, 9, 13, 14, 16]).
* This work was supported by the National Natural Science Foundation of China (Grant N. 11171243).
c© T. ZHAO, X. ZHANG, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2 187
188 T. ZHAO, X. ZHANG
In order to unify and generalize the related results, in this paper, we introduce a new kind of
generalized normal subgroup which can generalize both the SS-quasinormality and the S-embedded
property (and so it contains the s-permutability and c-normality) properly.
Definition 1.1. Let H be a subgroup of a finite group G, then H is said to be weakly SS-quasi-
normal in G, if there exists a normal subgroup T of G such that HT is s-permutable and H ∩ T is
SS-quasinormal in G.
Remark 1.1. From the definition, it is easy to see that every S-embedded subgroup and SS-
quasinormal subgroup of G is weakly SS-quasinormal in G. In general, a weakly SS-quasinormal
subgroup of G need not to be S-embedded or SS-quasinormal in G. For instance, we let G = S5 be
the symmetric group of degree 5.
Example 1.1. Let H = S4 and P ∈ Syl5(G). Since HP = PH = G, H is SS-quasinormal
and thus weakly SS-quasinormal in G. Since the only nontrivial normal subgroups of G are A5 and
G, but neither H nor H ∩A5 = A4 is s-permutable in G, H is not S-embedded in G.
Example 1.2. Let K = 〈(12)〉 and T = A5. Since T � G is a complement of K, K is weakly
SS-quasinormal in G. Since the only supplement of K to G are A5 and G itself, but K〈(12345)〉 6=
6= 〈(12345)〉K, K is not SS-quasinormal in G.
From some minimal subgroup’s normalities to characterize the structure of a finite group is an
active topic in the group theory. A number of meaningful results have been obtained under the
assumption that some minimal subgroups of G are well located. For example, Buckley [3] and Itô
(see [11], III, 5.3) have got some well-known results about the supersolublity and nilpotency of a
finite group, respectively. Since then, a series of papers have dealt with generalizations of the results
of Itô and Buckley by using the theory of formations and some generalized normal subgroups (see,
for example, [1, 2, 4, 10, 16]). In this paper, we investigate the influence of some weakly SS-qu-
asinormal minimal subgroups on the structures of a finite group G. Some new results about the
nilpotency of G are obtained, we also generalized some known ones.
2. Preliminaries. In this section, we list some basic results which will be useful in the sequel.
Lemma 2.1. Let H be an s-permutable subgroup of G.
(1) If K ≤ G, then H ∩K is s-permutable in K.
(2) If N �G, then HN/N is s-permutable in G/N .
(3) If H is a p-subgroup of G for some prime p, then NG(H) ≥ Op(G).
Proof. The proof of the statements can be seen in [12] and [5].
Lemma 2.2 ([13], Lemma 2.1) . Suppose that H is SS-quasinormal in a group G.
(1) If H ≤ K ≤ G, then H is SS-quasinormal in K.
(2) If N �G, then HN/N is SS-quasinormal in G/N .
Lemma 2.3 ([13], Lemma 2.2) . Let P be a p-subgroup of G, p a prime. Then P is s-permutable
in G if and only if P ≤ Op(G) and P is SS-quasinormal in G.
Now, we can prove that:
Lemma 2.4. Suppose that H is weakly SS-quasinormal in a group G, N �G.
(1) If H ≤ K ≤ G, then H is weakly SS-quasinormal in K.
(2) If N ≤ H, then H/N is weakly SS-quasinormal in G/N.
(3) Let π be a set of primes, H a π-subgroup and N a normal π′-subgroup of G. Then HN/N
is weakly SS-quasinormal in G/N.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
WEAKLY SS-QUASINORMAL MINIMAL SUBGROUPS AND THE NILPOTENCY OF A FINITE GROUP 189
(4) If H ≤ K � G, then G has a normal subgroup L contained in K such that HL is s-
permutable and H ∩ L is SS-quasinormal in G.
Proof. The statements (1), (2) and (4) can be deduced directly by Lemmas 2.1 and 2.2. Now
we prove the statement (3). By hypotheses, there exists a normal subgroup T of G such that HT is
s-permutable and H ∩ T is SS-quasinormal in G. It is easy to see that TN/N � G/N, by Lemma
2.1(2) we know (HN/N)(TN/N) = HTN/N is s-permutable in G/N . Since H is a π-group and
N a π′-group,
|H ∩ TN | = |H| · |TN |π
|HTN |π
=
|H| · |T |π
|HT |π
= |H ∩ T |.
This implies that H ∩ TN = H ∩ T . Hence (HN/N) ∩ (TN/N) = (HN ∩ TN)/N = (H ∩
∩ TN)N/N = (H ∩ T )N/N, which is SS-quasinormal in G/N by Lemma 2.2(2). Thus HN/N is
weakly SS-quasinormal in G/N, as required.
Lemma 2.4 is proved.
The following results is well known, one can see [21] (Lemma 2.2) for example.
Lemma 2.5. Let G be a group and p a prime divisor of |G| with (|G|, p− 1) = 1.
(1) If N is normal in G of order p, then N lies in Z(G).
(2) If G has cyclic Sylow p-subgroups, then G is p-nilpotent.
(3) If M is a subgroup of G with index p, then M is normal in G.
Lemma 2.6. Let F be a saturated formation containing the classes of all nilpotent groups N ,
H a normal subgroup of G. If G/H ∈ F and H ≤ Z(G), then G ∈ F .
Proof. Let f and F be the canonical definitions of N and F , respectively. Pick an chief factor
M/N of G contained in H, then M/N is a p-group for some prime p. Since M ≤ H ≤ Z(G),
M/N ≤ Z(G/N). Thus G/CG(M/N) = 1 ∈ f(p). Since N ⊆ F , f(p) ⊆ F (p) by [6] (IV,
Proposition 3.11). It follows that G/CG(M/N) ∈ F (p). The arbitrary choice of M/N implies that
there exists a normal chain of G contained in H such that every G-chief factor is F-central. Since
G/H ∈ F , it follows that G ∈ F .
Lemma 2.7 ([16], Lemma 2.8) . Suppose that P is a normal p-subgroup of G contained in
Z∞(G), then CG(P ) ≥ Op(G).
Lemma 2.8 ([11], X. 13) . Let F ∗(G) be the generalized Fitting subgroup of G.
(1) If M is a normal subgroup of G, then F ∗(M) ≤ F ∗(G).
(2) F ∗(G) 6= 1, if 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 soluble, then F ∗(G) = F (G).
(4) If K ≤ Z(G), then F ∗(G/K) = F ∗(G)/K.
3. Main results.
Theorem 3.1. Suppose that p is a prime divisor of a groupG with (|G|, p−1) = 1, P ∈ Sylp(G).
If every cyclic subgroup of P ∩GNp with prime order or order 4 (if p = 2 and P is non-abelian) not
having a p-nilpotent supplement in G is weakly SS-quasinormal in G, then G is a p-nilpotent group.
Proof. Suppose that the result is false and let G be a counterexample of minimal order. Then we
have
(1) Every proper subgroup of G is p-nilpotent, GNp = P is not a cyclic group.
LetM be a proper subgroup of G. SinceM/(M∩GNp) ∼= MGNp/GNp ≤ G/GNp is p-nilpotent,
MNp ≤ M ∩ GNp . Now, let Mp be a Sylow p-subgroup of M . Without loss of generality, we may
assume that Mp ≤ P and so Mp∩MNp ≤ P ∩GNp . By Lemma 2.4, we know every cyclic subgroup
of Mp ∩MNp with prime order or order 4 (if p = 2 and Mp is non-abelian) not having a p-nilpotent
supplement in M is weakly SS-quasinormal in M . Thus M satisfies the hypotheses of the theorem.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
190 T. ZHAO, X. ZHANG
The minimal choice of G implies that M is p-nilpotent and so G is a minimal non-p-nilpotent group.
By [11] (IV, Theorem 5.4), G has a normal Sylow p-subgroup P and a non-normal cyclic Sylow
q-subgroup Q such that G = PQ; P/Φ(P ) is a minimal normal subgroup of G/Φ(P ). Moreover, P
is of exponent p if p > 2 and exponent at most 4 if p = 2. On the other hand, the minimal choice of
G implies that GNp = P . By Lemma 2.5, we may also assume that P is not cyclic.
(2) Some minimal subgroup X/Φ(P ) of P/Φ(P ) is not s-permutable in G/Φ(P ).
If every minimal subgroup of P/Φ(P ) is s-permutable in G/Φ(P ), then by [17] (Lemma 2.11)
we know P/Φ(P ) has a maximal subgroup which is normal in G/Φ(P ). Since P/Φ(P ) is a chief
factor of G, |P/Φ(P )| = p and so P is cyclic, this contradicts with (1). Thus there exists some
minimal subgroup X/Φ(P ) of P/Φ(P ) such that X/Φ(P ) is not s-permutable in G/Φ(P ).
(3) 〈x〉 is weakly SS-quasinormal in G for any x ∈ X\Φ(P ).
Let x ∈ X\Φ(P ), then by (1) we know 〈x〉 is a cyclic group of order p or 4. Let T be any
supplement of 〈x〉 in G, then G = 〈x〉T and P = P ∩〈x〉T = 〈x〉(P ∩T ). Since P/Φ(P ) is abelian,
(P ∩T )Φ(P )/Φ(P )�G/Φ(P ) and hence (P ∩T )Φ(P )�G. Thus P ∩T ≤ Φ(P ) or P ∩T = P, as
P/Φ(P ) is a chief factor of G. If P ∩ T ≤ Φ(P ) for some supplement T of 〈x〉 in G, then P = 〈x〉
is cyclic, this contradicts with (1). Now assume that P ∩ T = P for any supplement T . Then T = G
is the unique supplement of 〈x〉 in G. Since G is not p-nilpotent, 〈x〉 is weakly SS-quasinormal in
G by the hypotheses.
(4) The final contradiction.
By (3) and Lemma 2.4(4), there exists a normal subgroup K of G contained in P such that 〈x〉K
is s-permutable and 〈x〉 ∩ K is SS-quasinormal in G. Since 〈x〉 ∩ K ≤ P = Op(G), 〈x〉 ∩ K is
s-permutable in G by Lemma 2.3. Since P/Φ(P ) is a chief factor of G, K ≤ Φ(P ) or K = P .
If K ≤ Φ(P ), then X/Φ(P ) = 〈x〉KΦ(P )/Φ(P ) is s-permutable in G/Φ(P ), a contradiction. If
K = P, then 〈x〉 = 〈x〉∩K is s-permutable in G and so X/Φ(P ) = 〈x〉Φ(P )/Φ(P ) is s-permutable
in G/Φ(P ), a contradiction too.
Theorem 3.1 is proved.
Next, by assuming that some minimal subgroups lie in the hypercenter of G and some cyclic
subgroups of order 4 having the weakly SS-quasinormal properties, we give out some criteria about
the nilpotency of a group G.
Theorem 3.2. Let E be a normal subgroup of G such that G/E is nilpotent. If every minimal
subgroup of E is contained in Z∞(G) and every cyclic subgroup of E with order 4 is weakly
SS-quasinormal in G or also lies in Z∞(G), then G is nilpotent.
Proof. Suppose that the result is false and let G be a counterexample of minimal order. Then we
have
(1) Every proper subgroup of G is nilpotent.
Let K be an arbitrary proper subgroup of G. Since G/E is nilpotent, K/K ∩ E ∼= KE/E is
nilpotent. Let H be a minimal subgroup of K ∩E, then H ≤ Z∞(G)∩K ≤ Z∞(K). For any cyclic
subgroup U of K∩E of order 4, by hypotheses U is weakly SS-quasinormal in G or lies in Z∞(G).
Then by Lemma 2.4, U is weakly SS-quasinormal in K or lies in Z∞(G) ∩ K ≤ Z∞(K). Thus
(K,K∩E) satisfies the hypotheses of the theorem in any case. The minimal choice of G implies that
K is nilpotent, thus G is a minimal non-nilpotent group. By [11] (II, Theorem 5.2), we can deduce
that G = PQ, where P is a normal Sylow p-subgroup and Q a non-normal cyclic Sylow q-subgroup
of G; P/Φ(P ) is a chief factor of G; exp(P ) = p or 4.
(2) p = 2, exp(P ) = 4 and every cyclic subgroup of P ≤ E with order 4 is weakly SS-quasi-
normal in G.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
WEAKLY SS-QUASINORMAL MINIMAL SUBGROUPS AND THE NILPOTENCY OF A FINITE GROUP 191
Since G/E is nilpotent and G/P ∩ E . G/P × G/E, G/P ∩ E is nilpotent. If P � E, then
P ∩E < P and Q(P ∩E) < G. Thus Q(P ∩E) is nilpotent by (1), then Q(P ∩E) = Q× (P ∩E)
and Q charQ(P ∩E). On the other hand, G/P ∩E = P/P ∩E×Q(P ∩E)/P ∩E, it follows that
Q(P ∩E)/P ∩E�G/P ∩E and Q(P ∩E)�G. Therefore, Q�G and G = P ×Q, a contradiction.
Thus we have P ≤ E. Since P is a normal Sylow p-subgroup of G, all elements of order p or 4 (if
p = 2) of G are contained in P and so contained in E. If p > 2 or p = 2 and every cyclic subgroup
of P with order 4 lies in Z∞(G), then by (1) and hypotheses, P ≤ Z∞(G). Therefore, Lemma 2.7
implies that G = PQ = P ×Q is nilpotent, a contradiction. Thus by hypotheses, we know that (2)
holds.
(3) Every x ∈ P\Φ(P ) is weakly SS-quasinormal in G.
If there exists some x ∈ P\Φ(P ) such that o(x) = 2, we denote M = 〈x〉G ≤ P, then
MΦ(P )/Φ(P )�G/Φ(P ). Since P/Φ(P ) is a minimal normal subgroup ofG/Φ(P ) andM * Φ(P ),
P = MΦ(P ) = M ≤ Z∞(G). Therefore, Lemma 2.7 implies that G = PQ = P ×Q is nilpotent, a
contradiction. Thus every x ∈ P\Φ(P ) is of order 4. By (2), we know 〈x〉 is weakly SS-quasinormal
in G.
(4) Some minimal subgroup of P/Φ(P ) is not s-permutable in G/Φ(P ).
If every minimal subgroup of P/Φ(P ) is s-permutable in G/Φ(P ), then by [17] (Lemma 2.11)
we know P/Φ(P ) has a maximal subgroup which is normal in G/Φ(P ). Since P/Φ(P ) is a chief
factor of G, |P/Φ(P )| = p. Since exp(P ) = 4, P is a cyclic group of order 4. Then Lemma 2.5
implies that Q�G and so G is nilpotent, a contradiction. Thus some minimal subgroup X/Φ(P ) of
P/Φ(P ) is not s-permutable in G/Φ(P ).
(5) The final contradiction.
Let x ∈ X\Φ(P ), then by (3) we know that x is of order 4 and 〈x〉 is weakly SS-quasinormal in
G. Thus there exists a normal subgroup K of G contained in P such that 〈x〉K is s-permutable and
〈x〉 ∩K is SS-quasinormal in G. Since 〈x〉 ∩K ≤ P = Op(G), by Lemma 2.3 we know 〈x〉 ∩K
is s-permutable in G. Since P/Φ(P ) is a chief factor of G, K ≤ Φ(P ) or K = P . If K ≤ Φ(P ),
then X/Φ(P ) = 〈x〉KΦ(P )/Φ(P ) is s-permutable in G/Φ(P ), a contradiction. If K = P, then
〈x〉 = 〈x〉∩K is s-permutable in G and so X/Φ(P ) = 〈x〉Φ(P )/Φ(P ) is s-permutable in G/Φ(P ),
a contradiction too.
Theorem 3.2 is proved.
Now, we can prove that:
Theorem 3.3. Let F be a saturated formation containing N . If every minimal subgroup of GF
lies in the F-hypercenter ZF (G) of G, then G ∈ F if and only if every cyclic subgroup of GF with
order 4 is weakly SS-quasinormal in G.
Proof. The necessity is obvious, we need to prove only the sufficiency.
Let 〈x〉 be a minimal subgroup of GF , then 〈x〉 ≤ ZF (G)∩GF which is contained in Z(GF ) by
[6] (IV, 6.10). From Lemma 2.4, we know that every cyclic subgroup of GF with order 4 is weakly
SS-quasinormal in GF . Theorem 3.2 implies that GF is nilpotent and so it is soluble. If GF ≤ Φ(G),
then G/Φ(G) ∈ F , hence G ∈ F . Thus we may assume that there exists a maximal subgroup M of
G such that G = MGF = MF (G). By [6] (IV, 1.17), we know MF ≤ GF . Hence every minimal
subgroup of MF is contained in ZF (G) ∩M ≤ ZF (M). By Lemma 2.4, every cyclic subgroup of
MF with order 4 is SS-quasinormal in M . Therefore, M satisfies the hypotheses of the theorem.
Then M ∈ F by induction. From [1] (Theorem 1 and Proposition 1), we know GF is a p-group
for some prime p ; GF/Φ(GF ) is a minimal normal subgroup of G/Φ(GF ); GF has exponent p if
p > 2 and exponent at most 4 if p = 2.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
192 T. ZHAO, X. ZHANG
If exp(GF ) = p, then GF = Ω1(G
F ) ≤ ZF (G) by the hypotheses, this would imply that
G ∈ F . Thus we may assume that p = 2 and exp(GF ) = 4. If there exists some x ∈ GF\Φ(GF )
such that o(x) = 2, denote H = 〈x〉G, then H � G and H ≤ Ω1(G
F ) ≤ ZF (G). On the other
hand, GF = HΦ(GF ) = H as GF/Φ(GF ) is a minimal normal subgroup of G/Φ(GF ). In this
case, G ∈ F . Next we assume that every x ∈ GF\Φ(GF ) is of order 4, and so by hypotheses 〈x〉 is
weakly SS-quasinormal in G. Let X/Φ(GF ) be an arbitrary minimal subgroup of GF/Φ(GF ) and
x ∈ X\Φ(GF ). Then there exists a normal subgroup K of G contained in GF such that 〈x〉K is
s-permutable and 〈x〉 ∩K is SS-quasinormal in G. Since 〈x〉 ∩K ≤ GF ≤ O2(G), 〈x〉 ∩K is s-
permutable in G by Lemma 2.3. Since GF/Φ(GF ) is a chief factor of G, K ≤ Φ(GF ) or K = GF .
If K ≤ Φ(GF ), then X/Φ(GF ) = 〈x〉KΦ(GF )/Φ(GF ) is s-permutable in G/Φ(GF ). If K = GF ,
then 〈x〉 = 〈x〉∩K is s-permutable in G and we also deduce that X/Φ(GF ) = 〈x〉Φ(GF )/Φ(GF ) is
s-permutable in G/Φ(GF ). This means that every minimal subgroup of GF/Φ(GF ) is s-permutable
in G/Φ(GF ). Then by [17] (Lemma 2.11) we know GF/Φ(GF ) has a maximal subgroup which is
normal in G/Φ(GF ). Since GF/Φ(GF ) is a chief factor of G, |GF/Φ(GF )| = 2. By Lemma 2.5,
we know GF/Φ(GF ) ≤ Z(G/Φ(GF )). Since (G/Φ(GF ))/(GF/Φ(GF )) ∼= G/GF ∈ F , Lemma
2.6 implies that G/Φ(GF ) ∈ F . Since Φ(GF ) ≤ Φ(G) and F is an saturated formation, G ∈ F , as
desired.
Theorem 3.3 is proved.
Theorem 3.4. A group G is nilpotent if and only if every minimal subgroup of F ∗(GN ) lies in
Z∞(G) and every cyclic subgroup of F ∗(GN ) with order 4 is weakly SS-quasinormal in G.
Proof. The necessity is obvious, we need to prove only the sufficiency. Suppose that the result is
false and let G be a counterexample of minimal order. Then
(1) Every proper normal subgroup of G is nilpotent.
Let M be a proper normal subgroup of G. Since M/(M ∩ GN ) ∼= MGN /GN ≤ G/GN is
nilpotent and MN �M ∩GN �GN , Lemma 2.8 implies that F ∗(MN ) ≤ F ∗(M ∩GN ) ≤ F ∗(GN ).
Moreover, M ∩ Z∞(G) ≤ Z∞(M). Now we can see easily that M satisfies the hypotheses of the
theorem. The minimal choice of G implies that M is nilpotent.
(2) F (G) is the unique maximal normal subgroup of G.
Let M be a maximal normal subgroup of G, then M is nilpotent by (1). Since the classes of all
nilpotent groups formed a Fitting class, the nilpotency of M implies that M = F (G) is the unique
maximal normal subgroup of G.
(3) GN = G = G
′
and F ∗(G) = F (G) < G.
If GN < G, then GN is nilpotent by (1). Thus, F ∗(GN ) = GN by Lemma 2.8. Now Theorem
3.2 implies immediately that G is nilpotent, a contradiction. Hence, we must have GN = G. Since
GN ≤ G′, it follows that G′ = G. Hence G/F (G) cannot be cyclic of prime order. Thus G/F (G)
is a non-abelian simple group. If F (G) < F ∗(G), then F ∗(GN ) = F ∗(G) = G by (2). Again by
Theorem 3.2, we can deduce that G is nilpotent, which is a contradiction.
(4) The final contradiction.
Since F (G) = F ∗(G) 6= 1, we may choose the smallest prime divisor p of |F (G)| such that
Op(G) 6= 1. Then for any Sylow q-subgroup Q of G (q 6= p), we consider the subgroup G0 =
= Op(G)Q. It is clear that GN0 ≤ Op(G) and G0 ∩ Z∞(G) ≤ Z∞(G0). Hence, every minimal
subgroup of GN0 lies in Z∞(G0) and every cyclic subgroup of GN0 with order 4 is weakly SS-
quasinormal in G0. By Theorem 3.2, we know G0 is nilpotent. Hence, G0 = Op(G) × Q and
Q ≤ CG(Op(G)). Consequently, G/CG(Op(G)) is a p-group. Thus we have CG(Op(G)) = G by (3),
namely Op(G) ≤ Z(G). Now we consider the factor group G = G/Op(G). First we have F ∗(G) =
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
WEAKLY SS-QUASINORMAL MINIMAL SUBGROUPS AND THE NILPOTENCY OF A FINITE GROUP 193
= F ∗(G)/Op(G) by Lemma 2.8(4). Besides that, for any element x of odd prime order in F ∗(G),
since Op(G) is the Sylow p-subgroup of F ∗(G), x can be viewed as the image of an element x of odd
prime order in F ∗(G). It follows that x lies in Z∞(G) and x lies in Z∞(G), as Z∞(G/Op(G)) =
= Z∞(G)/Op(G). This shows that G satisfies the hypotheses of the theorem. By the minimal choice
of G, we can conclude that G is nilpotent and so is G.
Theorem 3.4 is proved.
Now, we can get a more precise result:
Theorem 3.5. Let F be a saturated formation containing N . Then G ∈ F if and only if every
minimal subgroup of F ∗(GF ) lies in ZF (G) and every cyclic subgroup of F ∗(GF ) with order 4 is
weakly SS-quasinormal in G.
Proof. Only the sufficiency needs to be verified. By [6] (IV, 6.10), GF ∩ ZF (G) ≤ Z(GF ) ≤
≤ Z∞(GF ). Consequently, every minimal subgroup of F ∗(GF ) is contained in Z∞(GF ). By the hy-
potheses and Lemma 2.4, every cyclic subgroup of F ∗(GF ) with order 4 is weakly SS-quasinormal
in GF . By Theorem 3.4, we see that GF is nilpotent and so F ∗(GF ) = GF . Now by Theorem 3.3,
we can deduce that G ∈ F , as required.
Theorem 3.5 is proved.
4. Applications. Since all normal, quasinormal, s-permutable, c-normal, SS-quasinormal, nearly
s-normal [19] and S-embedded subgroups of G are weakly SS-quasinormal in G, our results have
many meaningful corollaries. Here, we list some of them.
Corollary 4.1 (see [20]) . G is 2-nilpotent if every cyclic subgroup of G with order 2 or order 4
is c-normal in G.
Corollary 4.2. Let p be a prime divisor of G with (|G|, p − 1) = 1, P ∈ Sylp(G). If every
cyclic subgroup of P ∩GNp with prime order or order 4 (if p = 2 and P is non-abelian) not having
a p-nilpotent supplement in G is SS-quasinormal (nearly s-normal, S-embedded) in G, then G is a
p-nilpotent group.
Corollary 4.3 (see [2]) . Let F be a saturated formation such that N ⊆ F . Let G be a group
such that every element of GF of order 4 is c-normal in G. Then G belongs to F if and only if 〈x〉
lies in the F-hypercenter ZF (G) of G for every element x ∈ GF of order 2.
Corollary 4.4 (see [15]) . Let F be a saturated formation containing N and let G be a group.
Then G ∈ F if and only if GF is solvable and every element of order 4 of F (GF ) is c-normal in G
and x lies in the F-hypercenter ZF (G) of G for every element x of prime order of F (GF ).
Corollary 4.5 (see [16]) . Suppose N is a normal subgroup of a group G such that G/N is
nilpotent. Suppose every minimal subgroup of N is contained in Z∞(G), every cyclic subgroup of
order 4 of N is s-permutable in G or lies also in Z∞(G), then G is nilpotent.
Corollary 4.6 (see [14]) . Let F be a saturated formation such that N ⊆ F , and let G be a
group. Every cyclic subgroup of order 4 of GF (or F ∗(GF )) is SS-quasinormal in G. Then G
belongs to F if and only if every subgroup of prime order of GF (or F ∗(GF )) lies in the F-
hypercenter ZF (G) of G.
Corollary 4.7. Let F be a saturated formation containing N . Then G ∈ F if and only if every
minimal subgroup of F ∗(GF ) lies in ZF (G) and every cyclic subgroup of F ∗(GF ) with order 4 is
nearly s-normal or S-embedded in G.
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Received 21.04.12,
after revision — 27.05.13
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
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| id | umjimathkievua-article-2122 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:19:07Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/5c/a290d5f08ad8272ca6602b71d94c5e5c.pdf |
| spelling | umjimathkievua-article-21222019-12-05T10:24:43Z Weakly SS-Quasinormal Minimal Subgroups and the Nilpotency of a Finite Group Слабко SS-квазінормальні мінімальш підгрупи та нільпотентність скінченної групи Zhang, Xirong Zhao, Tao Чжан, Хіронг Чжао, Тао A subgroup H is said to be an s-permutable subgroup of a finite group G provided that the equality HP =PH holds for every Sylow subgroup P of G. Moreover, H is called SS-quasinormal in G if there exists a supplement B of H to G such that H permutes with every Sylow subgroup of B. We show that H is weakly SS-quasinormal in G if there exists a normal subgroup T of G such that HT is s-permutable and H \ T is SS-quasinormal in G. We study the influence of some weakly SS-quasinormal minimal subgroups on the nilpotency of a finite group G. Numerous results known from the literature are unified and generalized. Підгрупа $H$ називається $s$-переставною підгрупою скінченної групи $G$ за умови, що $HP = PH$ виконується для кожної силовської підгрупи $P$ групи $G$; $H$ називається $SS$-квазінормальною в $G$, якщо існує доповнення B підгрупи $H$ до $G$ таке, що $H$ можна переставити з кожною силовською підгрупою $B$. Показано, що $H$ є слабко $SS$-квазінормальною в $G$, якщо існує нормальна підгрупа $T$ групи $G$ така, що $HT$ є $s$-переставною, а $H \ T$ є $SS$-квазінормальною в $G$. Досліджено вплив деяких слабко $SS$-квазінормальних мінімальних підгруп на нільпотентність скінченної групи $G$. Велику кількість відомих з літератури результатів упорядковано та узагальнено. Institute of Mathematics, NAS of Ukraine 2014-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2122 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 2 (2014); 187–194 Український математичний журнал; Том 66 № 2 (2014); 187–194 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2122/1246 https://umj.imath.kiev.ua/index.php/umj/article/view/2122/1247 Copyright (c) 2014 Zhang Xirong; Zhao Tao |
| spellingShingle | Zhang, Xirong Zhao, Tao Чжан, Хіронг Чжао, Тао Weakly SS-Quasinormal Minimal Subgroups and the Nilpotency of a Finite Group |
| title | Weakly SS-Quasinormal Minimal Subgroups and the Nilpotency of a Finite Group |
| title_alt | Слабко SS-квазінормальні мінімальш підгрупи та нільпотентність скінченної групи |
| title_full | Weakly SS-Quasinormal Minimal Subgroups and the Nilpotency of a Finite Group |
| title_fullStr | Weakly SS-Quasinormal Minimal Subgroups and the Nilpotency of a Finite Group |
| title_full_unstemmed | Weakly SS-Quasinormal Minimal Subgroups and the Nilpotency of a Finite Group |
| title_short | Weakly SS-Quasinormal Minimal Subgroups and the Nilpotency of a Finite Group |
| title_sort | weakly ss-quasinormal minimal subgroups and the nilpotency of a finite group |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2122 |
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