EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 3, 2021, 1002-1014
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Finite Groups With Minimal CSS-Subgroups

Abd El-Rahman Heliel1,2,∗, Rola Hijazi1, Shorouq Al-Shammari1,3

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589,
Saudi Arabia
2 Department of Mathematics and Computer Science, Faculty of Science,
Beni-Suef University, Beni-Suef 62511, Egypt
3 Department of Science and Technology, The University College in Al Khafji,
University of Hafr Al Batin, Al Khafji, Saudi Arabia

Abstract. Let G be a finite group. A subgroup H of G is called SS-quasinormal in G if there is
a supplement B of H to G such that H permutes with every Sylow subgroup of B. A subgroup H
of G is called CSS-subgroup in G if there exists a normal subgroup K of G such that G = HK
and H ∩K is SS-quasinormal in G. In this paper, we investigate the influence of minimal CSS-
subgroups of G on its structure. Our results improve and generalize several recent results in the
literature.

2020 Mathematics Subject Classifications: 20D10, 20D15, 20D20

Key Words and Phrases: CSS-subgroup, c-normal subgroup, SS-quasinormal subgroup, p-
nilpotent group, saturated formation.

1. Introduction

All groups considered in this paper are finite. The terminology and notions employed
agree with standard usage, as in [2, 5], and G always denotes a finite group.

Following Kegel [9], a subgroup H of G is said to be S-quasinormal in G if H permutes
with every Sylow subgroup of G, i.e, HP = PH for any Sylow subgroup P of G. A
subgroup H of G is said to be c-normal in G if G has a normal subgroup K such that
G = HK and H ∩ K 6 HG, where HG = CoreG(H) is the largest normal subgroup
of G contained in H (see Wang [18]). Recently, in 2008, Li et al. [12] extended S-
quasinormal subgroups of a group G to SS-quasinormal subgroups and they gave the
following definition: A subgroup H of G 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.

Obviously, every S-quasinormal subgroup is SS-quasinormal. The converse is not true
in general. For instance, S3 is SS-quasinormal subgroup of the symmetric group S4 but

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.4036

Email addresses: heliel9@yahoo.com; aahsalem1@kau.edu.sa (A. Heliel),
rhijazi@kau.edu.sa (R. Hijazi), salshammari0054@stu.kau.edu.sa (S. Al-Shammari)

http://www.ejpam.com 1002 c© 2021 EJPAM All rights reserved.



A. Heliel, R. Hijazi, S. Al-Shammari / Eur. J. Pure Appl. Math, 14 (3) (2021), 1002-1014 1003

not S-quasinormal. More recently, in 2019, Zhao et al. [26] introduced a new subgroup
embedding property of a finite group, called CSS-subgroup, which generalize and unify
both of c-normality and SS-quasinormality as follows: A subgroup H of G is called CSS-
subgroup of G if there exists a normal subgroup K of G such that G = HK and H ∩K is
SS-quasinormal in G. It is clear that each of c-normality and SS-quasinormality concepts
implies CSS-subgroup. The converse does not hold in general (see [26, Examples 1 and
2]).

Over years, many authors studied the influence of minimal subgroups of a finite group
on its structure (a subgroup of prime order is called a minimal subgroup). In this context,
Buckley [3] got the supersolvability of a group of odd order when all its minimal subgroups
are normal. In [17], Shaalan proved that a group G is supersolvable if all subgroups of
prime order p or of order 4 (if p = 2) of G are S-quasinormal in G. Later on, Wang [18] got
the same result of Shaalan [17] just he replaced S-quasinormality by c-normality. By using
the SS-quasinormality concept, Li et al. [11] extended these results through the theory
of formations and proved that: Let F be a saturated formation containing U and let G be
a group. Then G ∈ F if and only if G has a normal subgroup H such that G/H ∈ F and
every subgroup of F ∗(H) of prime order p or of order 4 (if p = 2) is SS-quasinormal in
G, where F ∗(H) is the generalized Fitting subgroup of H. Also, Wei et al. in [21] used
the c-normality concept and obtained the same previous result. For more results in this
direction (see [1, 11, 12, 16–18, 20, 21, 24]).

The main purpose of this paper is to improve and extend the above mentioned results
by using the recent concept CSS-subgroup. More precisely, we investigate the structure
of a finite group G when every subgroup of G of prime order p or of order 4 (if p = 2) is
CSS-subgroup in G.

2. Basic Definitions and Preliminaries

In this section, we list some definitions and state some known results from the literature
which will be used in proving our results.

A class of groups F is said to be a formation if F is closed under taking epimorphic
images and every group G has a smallest normal subgroup with quotient in F. This
subgroup is called the F-residual of G and it is denoted by GF. A formation F is called
saturated if it is closed under taking Frattini extensions. Throughout this paper, U and
N will denote the classes of supersolvable groups and nilpotent groups, respectively. It is
known that U and N are saturated formations (see [7, Satz 8.6, p. 713 and Satz 3.7, p.
270]).

A normal subgroup N of a group G is an F-hypercentral subgroup of G provided N
possesses a chain of subgroups 1 = N0EN1E...ENs = N such that Ni+1/Ni is an F-central
chief factor of G (see [5, p. 387]). The product of all F-hypercentral subgroups of G is
again an F-hypercentral subgroup, denoted by ZF(G), and it is called the F-hypercenter of
G (see [5, IV 6.8]). For the formation U, the U-hypercenter of a group G will be denoted
by ZU(G), that is, ZU(G) is the product of all normal subgroups N of G such that each
chief factor of G below N has prime order and for the formation N, the N-hypercenter of



A. Heliel, R. Hijazi, S. Al-Shammari / Eur. J. Pure Appl. Math, 14 (3) (2021), 1002-1014 1004

a group G is simply the terminal member Z∞(G) of the ascending central series of G. For
more details about saturated formations, see [5, IV].

For any group G, the generalized Fitting subgroup F ∗(G) is the set of all elements x
of G which induce an inner automorphism on every chief factor of G.

Lemma 1. (See [26, Lemma 2.3]) Let H be CSS-subgroup of G.

(1) If H 6M 6 G, then H is CSS-subgroup of M .

(2) Let N E G and N 6 H. Then H is CSS-subgroup of G if and only if H/N is
CSS-subgroup of G/N .

(3) Let π be a set of some primes and N a normal π′-subgroup of G. If H is a π-subgroup
of G, then HN/N is CSS-subgroup of G/N .

Lemma 2. (See [7, Satz 5.4, p. 434 and Satz 5.2, p. 281]) Let G be a minimal non p-
nilpotent group (a non p-nilpotent group all of its proper subgroups are p-nilpotent), where
p is a prime.

(1) G is a minimal non-nilpotent group.

(2) G = PQ, where P is a normal Sylow p-subgroup of G and Q is a non normal cyclic
Sylow q-subgroup of G.

(3) P/Φ(P ) is a minimal normal subgroup of G/Φ(P ).

(4) If p > 2, then the exponent of P is p and when p = 2, the exponent of P is at most
4.

Lemma 3. (See [17, Theorem 3.2]) Let p be the smallest prime dividing |G| and P a
Sylow p-subgroup of G. If every subgroup of P of order p or of order 4 (if p = 2) is
S-quasinormal in G, then G is p-nilpotent.

Lemma 4. (See [23]) Let H be a subnormal subgroup of G.

(1) If H is a Hall-subgroup of G, then H is normal in G.

(2) If H is a π-subgroup of G, then H 6 Oπ(G).

Lemma 5. (See [11, Lemma 2.2]) Suppose that P is a p-subgroup of G. Then P is
S-quasinormal in G if and only if P 6 Op(G) and P is SS-quasinormal in G.

Lemma 6. (See [13, Theorem 3.3]) Suppose that P is a normal p-subgroup of G, where
p > 2. If every subgroup of P of order p is S-quasinormal in G, then P 6 ZU(G).

Lemma 7. (See [22, Theorem 7.7, p. 31]) Let N be a normal subgroup of G such that
N 6 ZU(G). Then ZU(G/N) = ZU(G)/N .

Lemma 8. (See [22, Theorem 6.3, p. 220 and Corollary 7.8, p. 33]) Let P be a normal
p-subgroup of G such that |G/CG(P )| is a power of p. Then P 6 ZU(G).



A. Heliel, R. Hijazi, S. Al-Shammari / Eur. J. Pure Appl. Math, 14 (3) (2021), 1002-1014 1005

Lemma 9. (See [5, Propositin 3.11, p. 362]) If F1 and F2 are two saturated formations
such that F1 ⊆ F2, then ZF1(G) ⊆ ZF2(G).

Lemma 10. (See [4]) Let K be a normal subgroup of G such that G/K ∈ F, where F
is a saturated formation. If Ω(P ) 6 ZF(G), where P is a Sylow p-subgroup of K, then
G/Op′(K) ∈ F.

Lemma 11. (See [8, X 13] and [14, Lemma 2.3(4)]) Let M be a subgroup of G.

(1) If M is normal in G, then F ∗(M) 6 F ∗(G).

(2) F ∗(G) 6= 1 if G 6= 1.

(3) If F ∗(G) is solvable, then F ∗(G) = F (G).

(4) Suppose K is a subgroup of G contained in Z(G). Then F ∗(G/K) = F ∗(G)/K.

Lemma 12. (See [10, Corollary 3]) Let F be a saturated formation and G a group. Suppose
that CG(N) 6 N E G. Then G ∈ F if every cyclic subgroup of N of prime order or of
order 4 is contained in ZF(G).

Lemma 13. (See [15, Lemma 2.8]) Suppose that G is a group and P is a normal p-subgroup
of G contained in Z∞(G). Then CG(P ) > Op(G).

Lemma 14. (See [7, Satz 2.8, p. 420]) If P is a cyclic Sylow p-subgroup of G, where p is
the smallest prime dividing |G|, then G is p-nilpotent.

Lemma 15. (See [6, Theorem 3.10, p. 184]) If H is a p′-group of automorphisms of the
p-group P with p odd which acts trivially on Ω1(P ), then H = 1.

Lemma 16. (See [6, Theorem 2.4, p. 178]) If H is a p′-group of automorphisms of the
abelian p-group P which acts trivially on Ω1(P ), then H = 1.

3. Main Results

First we prove:

Theorem 1. Let p be the smallest prime dividing |G| and P a Sylow p-subgroup of G. If
every subgroup of P of prime order p or of order 4 (if p = 2) is CSS-subgroup of G, then
G is p-nilpotent.

Proof. Assume that the result is false and let G be a counterexample of minimal order.
Let L be an arbitrary proper subgroup of G. Then every subgroup of L of prime order p or
of order 4 (if p = 2) is CSS-subgroup of G by the hypothesis. Thus, by Lemma 11, every
subgroup of L of prime order p or of order 4 (if p = 2) is CSS-subgroup of L.

That means L satisfies the hypothesis of the theorem and so L is p-nilpotent by the minimal
choice of G. Hence, G is not p-nilpotent but all of its proper subgroups are p-nilpotent.



A. Heliel, R. Hijazi, S. Al-Shammari / Eur. J. Pure Appl. Math, 14 (3) (2021), 1002-1014 1006

By Lemma 2, G is a minimal non-nilpotent group and so G = PQ, where P is a normal
Sylow p-subgroup of G and Q is a non-normal cyclic Sylow q-subgroup of G, for some
prime q 6= p. Furthermore, if p > 2, then P is of exponent p and if p = 2, P is of exponent
at most 4. If every subgroup of P with order p or 4 (if p = 2) is S-quasinormal in G,
then, by Lemma 3, we get the p-nilpotency of G, a contradiction. Therefore, there exists a
subgroup S of P of prime order p or of order 4 (if p = 2) such that S is not S-quasinormal
in G. By hypothesis, S is CSS-subgroup of G. Then there exists a normal subgroup K
of G such that G = SK and S ∩ K is SS-quasinormal in G. Assume that K = G. It
follows that S is SS-quasinormal in G. Since P is normal in G, then S is subnormal in
G. Thus, by Lemma 4, S 6 Op(G). Applying Lemma 5, we get S is S-quasinormal in G,
a contradiction. Hence, K is a proper normal nilpotent subgroup of G which implies that
Q is characteristic in K. Therefore Q is a normal subgroup in G, a final contradiction
completing the proof.

Lemma 17. Let P be a non-trivial normal p-subgroup of G (where p > 2). If every
minimal subgroup of P is CSS-subgroup of G, then P 6 ZU(G).

Proof. We prove the theorem by induction on |G| + |P |. If every minimal subgroup
of P is S-quasinormal in G, then by Lemma 6, we get P 6 ZU(G) and we are done.
Thus, we may assume that P has a minimal subgroup L such that L is not S-quasinormal
in G. By the hypothesis of the lemma, L is CSS-subgroup of G, i.e., G has a normal
subgroup K such that G = LK and L ∩ K is SS-quasinormal in G. If L ∩ K 6= 1, we
have L ∩ K = L. Hence, L is SS-quasinormal in G. Since P is normal in G, then
L is subnormal in G. Lemma 4 implies that L 6 Op(G). Applying Lemma 5, L is
S-quasinormal in G, a contradiction. Therefore, we may assume L ∩ K = 1. Then,
P = P ∩ G = P ∩ LK = L(P ∩ K) and P ∩ K E G. By the hypothesis, every minimal
subgroup of the non-trivial normal p-subgroup P ∩ K is CSS-subgroup of G. This leads
to P ∩ K 6 ZU(G) by induction on |G| + |P |. Hence, P/(P ∩ K) 6 ZU(G/(P ∩ K)) as
P/(P ∩ K) is a normal subgroup of G/(P ∩ K) of order p. But P ∩ K 6 ZU(G), then
ZU(G/(P ∩ K)) = ZU(G)/(P ∩ K) by Lemma 7. Thus, P/(P ∩ K) 6 ZU(G)/(P ∩ K).
Now it follows easily that P 6 ZU(G).

Immediate consequence of Lemma 17 and Theorem 1, we have the following corollary:

Corollary 1. Let P be a normal p-subgroup of G. If every subgroup of P of prime order
p or of order 4 (if p = 2) is CSS-subgroup of G, then P 6 ZU(G).

Proof. Assume that p > 2. Then, by Lemma 17, P 6 ZU(G) and we are done. Hence,
consider p = 2. Let Q be any Sylow q-subgroup of G, where q 6= 2. It is clear that PQ is
a subgroup of G. Since every subgroup of P of prime order p or of order 4 (if p = 2) is
CSS-subgroup of G, then by Lemma 11, every subgroup of P of prime order p or of order
4 (if p = 2) is CSS-subgroup of PQ.

By applying Theorem 1, we have PQ is 2-nilpotent. This implies that PQ = P × Q
and so Q centralizes P . Thus, Op(G) 6 CG(P ) and it follows that |G/CG(P )| is a power
of 2. By Lemma 8, we conclude P 6 ZU(G).



A. Heliel, R. Hijazi, S. Al-Shammari / Eur. J. Pure Appl. Math, 14 (3) (2021), 1002-1014 1007

We now prove:

Theorem 2. Let F be a saturated formation containing U and G a group. Then G ∈ F if
and only if there exists a normal subgroup H in G such that G/H ∈ F and every subgroup
of H of prime order p or of order 4 (if p = 2) is CSS-subgroup of G.

Proof. If G ∈ F, then we set H = 1 and the result follows. Conversely, assume that
the result is false and let G be a counterexample of minimal order. By using Lemma 11
and repeated applications of Theorem 1, the group H has a Sylow tower of supersolvable
type which means that H has a normal Sylow p-subgroup P , where p is the largest prime
dividing |H|. Clearly, P is normal in G and hence (G/P )/(H/P ) ∼= G/H ∈ F. By Lemma
12, every subgroup of H/P of prime order or of order 4 (if p = 2) is CSS-subgroup of
G/P . Then, by the minimal choice of G, we have G/P ∈ F and so 1 6= GF 6 P . By the
hypothesis, every subgroup of GF of prime order p or of order 4 (if p = 2) is CSS-subgroup
of G. Then, by Corollary 1, GF 6 ZU(G). Since Ω(GF) 6 GF and ZU(G) 6 ZF(G), by
Lemma 9, we have Ω(GF) 6 GF 6 ZU(G) 6 ZF(G). Hence, Ω(GF) 6 ZF(G). Therefore,
by Lemma 10, G ∈ F, a contradiction.

The following corollaries are immediate consequences of Theorem 2:

Corollary 2. Let H be a normal subgroup of G such that G/H is supersolvable. If every
subgroup of H of prime order p or of order 4 (if p = 2) is CSS-subgroup of G, then G is
supersolvable.

Corollary 3. Let H be a normal subgroup of G such that (G/H)′ is nilpotent. If every
subgroup of H of prime order p or of order 4 (if p = 2) is CSS-subgroup of G, then G′ is
nilpotent.

Corollary 4. Let G be a group such that every subgroup of G of prime order p or of order
4 (if p = 2) is CSS-subgroup of G, then G is supersolvable.

Now we can prove:

Theorem 3. Let F be a saturated formation containing U and G a group. Then G ∈ F if
and only if G has a normal subgroup H such that G/H ∈ F and every subgroup of F ∗(H)
of prime order p or of order 4 (if p = 2) is CSS-subgroup of G.

Proof. If G ∈ F, then we set H = 1 and the theorem follows. Now we prove the
converse. By the hypothesis and Lemma 11, every subgroup of F ∗(H) of prime order p
or of order 4 (if p = 2) is CSS-subgroup of F ∗(H). Corollary 4 implies that F ∗(H) is
supersolvable. Hence, by Lemma 11, F ∗(H) = F (H). Then, by Corollary 1, Op(H) 6
ZU(G). Since ZU(G) 6 ZF(G), by Lemma 9, it follows that Op(H) 6 ZF(G) and so
F ∗(H) = F (H) 6 ZF(G). Applying Lemma 12, we get G ∈ F.

Immediately from Theorem 3, we have the following corollaries:

Corollary 5. Let H be a normal subgroup G such that G/H is supersolvable. If every
subgroup of F ∗(H) of prime order p or of order 4 (if p = 2) is CSS-subgroup of G, then
G is supersolvable.



A. Heliel, R. Hijazi, S. Al-Shammari / Eur. J. Pure Appl. Math, 14 (3) (2021), 1002-1014 1008

Corollary 6. If every subgroup of F ∗(G) of prime order p or of order 4 (if p = 2) is
CSS-subgroup of G, then G is supersolvable.

Corollary 7. Let F be a saturated formation containing U and G a group. Then G ∈ F if
and only if G has a solvable normal subgroup H such that G/H ∈ F and every subgroup
of F (H) of prime order p or of order 4 (if p = 2) is CSS-subgroup of G.

We now prove:

Theorem 4. Let G be a group. If every subgroup of G of prime order is contained in
Z∞(G) and every cyclic subgroup of order 4 of G is CSS-subgroup of G or lies in Z∞(G),
then G is nilpotent.

Proof. Assume that the result is false and let G be a counterexample of minimal order.
Let L be an arbitrary proper subgroup of G and K a cyclic subgroup of L of prime order
or of order 4. Then K 6 Z∞(G) ∩ L 6 Z∞(L). By hypotheses and Lemma 11, K is
CSS-subgroup of L. The minimal choice of G implies that L is nilpotent. Since L is an
arbitrary proper subgroup of G, we have that G is a minimal non-nilpotent group. Hence,
by Lemma 2, G = PQ, where P is a normal Sylow p-subgroup of G and Q is a non normal
cyclic Sylow q-subgroup of G, p 6= q. Moreover, P/Φ(P ) is a minimal normal subgroup of
G/Φ(P ). Now we have:

(1) p = 2 and every element of order 4 is CSS-subgroup of G.
Assume that p > 2. By Lemma 2, the exponent of P is p. Then, by the hy-

potheses, P 6 Z∞(G). Applying Lemma 13, Op(G) 6 CG(P ) which means that
G = PQ = P ×Q is nilpotent, a contradiction. If every element of order 4 of G lies
in Z∞(G), then P 6 Z∞(G) which means that G = PQ = P ×Q is nilpotent, again
contradiction.

(2) For every x ∈ P \ Φ(P ), |x| = 4.
Assume that |x| 6= 4. Then there exists x ∈ P \Φ(P ) and |x| = 2. Since P EG, we

have that < xG >6 P . Then < xG > Φ(P )/Φ(P ) EG/Φ(P ). But as we mentioned
above P/Φ(P ) is a minimal normal subgroup of G/Φ(P ). Then P =< xG > Φ(P ) =
< xG >6 Z∞(G). In particular; G is nilpotent, a contradiction.

(3) Finishing the proof.
From 2, every element x in P \ Φ(P ) is of order 4. From 1, < x > is CSS-

subgroup of G. Then there exists a normal subgroup S of G such that G =<
x > S and < x > ∩S is SS-quasinormal in G. Clearly, P ∩ S E G. Hence,
(P∩S)Φ(P )/Φ(P )EG/Φ(P ). Since P/Φ(P ) is a minimal normal subgroup G/Φ(P ),
it follows that either P ∩ S 6 Φ(P ) or P ∩ S = P . Assume first that P ∩ S 6 Φ(P ).
Then P = P ∩ G = P ∩ (< x > S) =< x > (P ∩ S) =< x > Φ(P ). Therefore,
P =< x > and this means that P is a cyclic normal Sylow 2-subgroup of G of order 4.
By Lemma 14, G is 2-nilpotent and so G = PQ = P×Q is nilpotent, a contradiction.
Thus, assume that P ∩ S = P . Then < x >=< x > ∩P =< x > ∩(P ∩ S) = (<
x > ∩P ) ∩ S =< x > ∩S. Hence, < x > is SS-quasinormal in G. < x >6 P E G



A. Heliel, R. Hijazi, S. Al-Shammari / Eur. J. Pure Appl. Math, 14 (3) (2021), 1002-1014 1009

implies that < x > is subnormal in G. By Lemma 4, < x >6 O2(G). Applying
Lemma 5, < x > is S-quasinormal in G. Thus, < x > Q 6 G. If < x > Q = G, then
< x >= P which implies G is nilpotent, a contradiction. Therefore, < x > Q < G
and it follows that < x > Q is nilpotent. Then < x > Q =< x > ×Q. Thus,
< x >6 NG(Q) implies that P 6 NG(Q) and so G = PQ = P × Q is nilpotent, a
final contradiction completing the proof.

Theorem 5. Let H be a normal subgroup of G such that G/H is nilpotent. If every
subgroup of H of prime order is contained in Z∞(G) and every cyclic subgroup of order 4
of H is CSS-subgroup of G or lies in Z∞(G), then G is nilpotent.

Proof. Assume that the result is false and let G be a counterexample of minimal
order. Let L be an arbitrary proper subgroup of G. Since G/H is nilpotent, we have
L/L ∩ H ∼= LH/H is nilpotent. The element of prime order or of order 4 of L ∩ H is
contained in Z∞(G)∩L 6 Z∞(L). By hypotheses and Lemma 11, every cyclic subgroup of
order 4 of L∩H is CSS-subgroup in L. Thus the pair (L,L∩H) satisfies the hypotheses
of the theorem in any case. Then L is nilpotent, that is, G is a minimal non-nilpotent
group. Applying Lemma 2, G = PQ, where P is normal Sylow p-subgroup of G and Q is
non normal cyclic Sylow q-subgroup of G, p 6= q. Since G/H and G/P are nilpotent, then
G/P ∩H 6 G/P × G/H is nilpotent. Now we deal with:

(1) P 6 H.
Assume that p > 2. Then, by Lemma 2, the exponent of P is p and so P =

P ∩ H 6 Z∞(G). Applying Lemma 13, we have Op(G) 6 CG(P ). This implies
G = PQ = P × Q is nilpotent, a contradiction. Thus, we may assume that p = 2.
Since P E G, it follows that every element of order 2 or 4 of G is contained in P ;
in particular in H. Thus, every element of order 2 of G lies in Z∞(G) and, by
hypotheses, every cyclic subgroup of order 4 is CSS-subgroup of G or lies also in
Z∞(G). Applying similar arguments to those in (2) and (3) of the proof of Theorem
4, we have that G is nilpotent, a contradiction.

(2) P 
 H.
Then P ∩ H < P and hence Q(P ∩ H) < G. Therefore, Q(P ∩ H) is nilpotent

which implies that Q(P ∩ H) = Q × (P ∩ H). Moreover, Q is characteristic in
Q(P ∩H). Clearly, as G/P ∩H = (P/P ∩H)(Q(P ∩H)/P ∩H) is nilpotent, then
Q(P ∩H)/P ∩H EG/P ∩H. Thus Q(P ∩H) EG. Hence QEG, a contradiction.

Theorem 6. Let H be a normal subgroup of G such that G/H is nilpotent and every
cyclic subgroup of order 4 of F ∗(H) is CSS-subgroup of G. Then G is nilpotent if and
only if every subgroup of prime order of F ∗(H) is contained in Z∞(G).

Proof. If G is nilpotent, then we set H = 1 and the result follows. Conversely, assume
that the result is false and let G be a counterexample of minimal order. With the same
arguments to those in steps (1) and (2) of the proof of Theorem 4.4 in [11], we have:



A. Heliel, R. Hijazi, S. Al-Shammari / Eur. J. Pure Appl. Math, 14 (3) (2021), 1002-1014 1010

(1) Every proper normal subgroup of G is nilpotent, and F (G) is the unique maximal
normal subgroup of G.

(2) H = G, G′ = G and F ∗(G) = F (G) < G.

(3) Let q be a minimal prime divisor of |F (G)| and Q a Sylow q-subgroup of F (G).
Then G/CG(Q) is a q-group.
Since F ∗(G) 6= 1, then we may assume that q is a minimal prime divisor of |F (G)|

and Q is a Sylow q-subgroup of F (G) which is a non-trivial normal subgroup of
G. Clearly, from hypotheses, Ω1(Q) 6 Z∞(G). Thus, by Lemma 13, CG(Ω1(Q)) >
Oq(G). If q > 2, then, by Lemma 15, CG(Q) > Oq(G). This implies that G/CG(Q)
is a q-group. If q = 2, let < x > be an arbitrary cyclic subgroup of Q of order
4. By hypotheses, < x > is CSS-subgroup of G. Then, there exists a normal
subgroup L of G such that G =< x > L and < x > ∩L is SS-quasinormal in
G. If < x > ∩L = 1, then L is a proper normal subgroup of G and, by (1), L is
nilpotent. It follows that any Sylow p-subgroup of L is normal in G, where p is any
prime number such that p 6= 2. Therefore, G is nilpotent, a contradiction. Hence
we may assume that < x >6 L and < x > is SS-quasinormal in G. Since Q is a
normal subgroup of G, it follows that < x > is subnormal in G. Hence, by Lemma
4, < x >6 O2(G). Applying Lemma 5, < x > is S-quasinormal in G. Now, let P
be any Sylow p-subgroup of G, where p 6= 2. Therefore < x > P 6 G. Clearly, as
< x > is subnormal in < x > P and < x > is a Sylow 2-subgroup of < x > P , we
have < x > is normal in < x > P . Hence, by Lemma 16, < x > P is nilpotent.
It follows that P 6 CG(< x >) and so O2(G) 6 CG(< x >). This implies that
O2(G) 6 CG(Q) and so G/CG(Q) is a 2-group.

(4) We have a contradiction.
By 2, G = G′ and so CG(Q) = G, Q 6 Z(G). By Lemma 11, F ∗(G/Q) = F ∗(G)/Q.

Let G = G/Q. Then, 3 imply that each element y of prime order n in F ∗(G) can be
viewed as an image in element y of prime order n in F ∗(G), For each n > q. Thus,
by hypotheses, y 6 Z∞(G). Since Q 6 Z(G), then Z∞(G/Q) = Z∞(G)/Q. Hence
y 6 Z∞(G/Q). Clearly, F ∗(G/Q) does not have an element of order 2. This means
that G satisfies the hypotheses of the theorem. Then G = G/Q is nilpotent by our
choice of G and so G is nilpotent which yields the desired contradiction.

4. Some Applications

As it was mentioned in the introduction each of c-normality and SS-quasinormality
subgroups implies CSS-subgroups. Therefore the following results are direct consequences
of our results.

Corollary 8. ([1, Lemma 3.1]) Let p be the smallest prime dividing |G| and P a Sylow
p-subgroup of a group G. If every subgroup of P of prime order or of order 4 (if p = 2) is
c-normal in G, then G is p-nilpotent.



A. Heliel, R. Hijazi, S. Al-Shammari / Eur. J. Pure Appl. Math, 14 (3) (2021), 1002-1014 1011

Corollary 9. ([18, Theorem 4.2]) Let G be a group such that every subgroup of G of prime
order or of order 4 (if p = 2) is c-normal in G, then G is supersolvable.

Corollary 10. ([1, Theorem 3.2] and [16, Theorem 3.9]) Let F be a saturated formation
containing U and G a group. G ∈ F if and only if there exists a normal subgroup H in G
such that G/H ∈ F and every subgroup of H of prime order or of order 4 (if p = 2) is
c-normal in G.

Corollary 11. ([1, Theorem 3.6] and [24, Theorem 3]) Let F be a saturated formation
containing U and G a group. G ∈ F if and only if there exists a normal solvable subgroup
H in G such that G/H ∈ F and every subgroup of F (H) of prime order or of order 4 (if
p = 2) is c-normal in G.

Corollary 12. ([21, Theorem 3.2]) Let F be a saturated formation containing U and G a
group. If G has a normal subgroup H such that G/H ∈ F and every subgroup of F ∗(H)
of prime order or of order 4 is c-normal in G, then G ∈ F.

Corollary 13. ([19, Theorem 3.1]) Let H be a normal subgroup of a group G such that
G/H is nilpotent and every cyclic subgroup of order 4 of F ∗(H) is c-normal in G, then
G is nilpotent if and only if every subgroup of prime order of F ∗(H) is contained in the
hypercenter Z∞(G) of G.

Corollary 14. Let p be the smallest prime dividing |G| and P a Sylow p-subgroup of a
group G. If every subgroup of P of prime order or of order 4 (if p = 2) is SS-quasinormal
in G, then G is p-nilpotent.

Corollary 15. ([11, Theorem 3.4]) Let G be a group such that every subgroup of G of
prime order or of order 4 (if p = 2) is SS-quasinormal in G, then G is supersolvable.

Corollary 16. Let F be a saturated formation containing U and G a group. G ∈ F if and
only if there exists a normal subgroup H in G such that G/H ∈ F and every subgroup of
H of prime order or of order 4 (if p = 2) is SS-quasinormal in G.

Corollary 17. ([11, Theorem 3.5]) Let F be a saturated formation containing U and G
a group. G ∈ F if and only if there exists a normal solvable subgroup H in G such
that G/H ∈ F and every subgroup of F (H) of prime order or of order 4 (if p = 2) is
SS-quasinormal in G.

Corollary 18. ([11, Theorem 3.6]) Let G be a group. If G has a normal subgroup H such
that G/H is supersolvable and every subgroup of F ∗(H) of prime order or of order 4 is
SS-quasinormal in G, then G is supersolvable.

Corollary 19. ([11, Theorem 3.7]) Let F be a saturated formation containing U and G a
group. Then G ∈ F if and only if G has a normal subgroup H such that G/H ∈ F and
every subgroup of F ∗(H) of prime order or of order 4 is SS-quasinormal in G.

Corollary 20. ([11, Theorem 4.1]) Let G be a group. If every subgroup of G of prime
order is contained in Z∞(G) and every cyclic subgroup of order 4 of G is SS-quasinormal
in G or lies in Z∞(G), then G is nilpotent.



REFERENCES 1012

Corollary 21. ([11, Theorem 4.2]) Let H be a normal subgroup of a group G such that
G/H is nilpotent. If every subgroup of H of prime order is contained in Z∞(G) and every
cyclic subgroup of order 4 of H is SS-quasinormal in G or lies in Z∞(G), then G is
nilpotent.

Corollary 22. ([11, Theorem 4.4]) Let H be a normal subgroup of a group G such that
G/H is nilpotent and every cyclic subgroup of order 4 of F ∗(H) is SS-quasinormal in G,
then G is nilpotent if and only if every subgroup of prime order of F ∗(H) is contained in
the hypercenter Z∞(G) of G.

Based on the results that have been achieved in this paper and [25, 26], the following
questions arise:

Question 1. Let P be a Sylow p-subgroup of a group G, where p is the smallest prime
dividing |G|. Assume that all maximal subgroups of P are CSS-subgroups of G. Is G
p-nilpotent?

Question 2. Assume that all maximal subgroups of every Sylow subgroup of a group G
are CSS-subgroups of G. Is G supersolvable?

Question 3. Let F be a saturated formation containing U and H a normal subgroup of G
such that G/H ∈ F. Assume that every non-cyclic Sylow subgroup P of H has a subgroup
D with 1 < |D| < |P | such that every subgroup of P of order |D| (and 4 if |D| = 2) is
CSS-subgroup of G. Is G ∈ F?

Question 4. Let F be a saturated formation containing U and H a normal subgroup of
G such that G/H ∈ F. Assume that every non-cyclic Sylow subgroup P of F ∗(H) has
a subgroup D with 1 < |D| < |P | such that every subgroup of P of order |D| (and 4 if
|D| = 2) is CSS-subgroup of G. Is G ∈ F?

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