EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 2, 2019, 571-576
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Finite Groups With Certain Permutability Criteria

Rola A. Hijazi1, Fatme M. Charaf1,∗

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah,
Saudi Arabia

Abstract. Let G be a finite group. A subgroup H of G is said to be S-permutable in G if it
permutes with all Sylow subgroups of G. In this note we prove that if P , the Sylow p-subgroup of
G (p > 2), has a subgroup D such that 1 < |D| < |P | and all subgroups H of P with |H| = |D|
are S-permutable in G, then G′ is p-nilpotent.

2010 Mathematics Subject Classifications: 20D10, 20D20

Key Words and Phrases: S -Permutable Subgroup, p-Nilpotent Group, Solvable Group, Super-
solvable Group.

1. Introduction

Throughout this note, G denotes a finite group. The relationship between the proper-
ties of the Sylow subgroups of a group G and its structure has been investigated by many
authors. Starting from Gaschűtz and Itő ([10], Satz 5.7, p.436) who proved that a group G
is solvable if all its minimal subgroups are normal. In 1970, Buckely [4] proved that a group
of odd order is supersolvable if all its minimal subgroups are normal (a subgroup of prime
order is called a minimal subgroup). Recall that a subgroup is said to be S-permutable
in G if it permutes with all Sylow subgroup of G. This concept, as a generalization of
normality, was introduced by Kegel [11] in 1962 and has been studied extensively in many
notes. For example, Srinivasan [15] in 1980 obtained the supersolvability of G under the
assumption that the maximal subgroups of all Sylow subgroups are S-permutable in G.
In 2000, Ballester-Bolinches et al. [3] introduced the c-supplementation concept of a finite
group: A subgroup H of a group G is said to be c-supplemented in G if there exists a
subgroup K of G such that G = HK and H ∩K ≤ HG, where HG = CoreG(H) is the
largest normal subgroup of G contained in H. By using this concept they were able to
prove that a group G is solvable if and only if every Sylow subgroup of G is c-supplemented
in G. Moreover, as an application, they got the supersolvability of a group G if all its
minimal subgroups and the cyclic subgroups of order 4 are c-supplemented in G.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i2.3399

Email addresses: Rhijazi@kau.edu.sa (R. Hijazi), fa-sharaf@hotmail.com (F. Charaf)

http://www.ejpam.com 571 c© 2019 EJPAM All rights reserved.



R. Hijazi, F. Charaf / Eur. J. Pure Appl. Math, 12 (2) (2019), 571-576 572

In 2014, Heliel [8] proved that G is solvable if each subgroup of prime odd order of
G is c-supplemented in G. Also he proved that G is solvable if and only if every Sylow
subgroup of odd order of G is c-supplemented in G. This improved and generalized the
results of Hall [6, 7], Ballester-Bolinches and Guo [2], and Ballester-Bolinches et al. [3].
Heliel also posted the following conjecture:

Let G be a finite group such that every non-cyclic Sylow subgroup P of odd order of
G has a subgroup D such that 1 < |D| ≤ |P | and all subgroups H of P with |H| = |D|
are c-supplemented in G. Is G solvable?

In the same year, Li et al. [12] presented a counterexample to show that the answer of
this conjecture is negative in general and then gave a generalization of Heliel’s theorems.

Example 1. Let G = A5 ×H, where A5 is the alternating group of degree 5 and H is
an elementary group of order pn with p > 5 and n ≥ 2. Then G satisfies the condition of
the preceding conjecture, but G is not solvable.

In 2015, Hijazi [9] continued the above mentioned investigations and proved the follow-
ing: Suppose that each Sylow subgroup P of G has a subgroup D such that 1 < |D| < |P |
and all subgroups H of P with |H| = |D| are S-permutable in G. Then G is solvable.

The main goal of this note is to prove the following main theorem:

Main Theorem 1. Let P be a Sylow p-subgroup of G (p > 2). Suppose that P has
a subgroup D such that 1 < |D| < |P | and all subgroups H of P with |H| = |D| are
S-permutable in G. Then G′ is p-nilpotent.

As immediate consequences of the main theorem we have:

Corollary 1. Let P be a Sylow p-subgroup of G (p > 2). Suppose that P has a subgroup
D such that 1 < |D| < |P | and all subgroups H of P with |H| = |D| are permutable in G.
Then G′ is p-nilpotent.

Corollary 2 ([9], Theorem 3.1). Suppose that each Sylow subgroup P of G has a subgroup
D such that 1 < |D| < |P | and all subgroups H of P with |H| = |D| are S-permutable in
G. Then G is solvable.

Corollary 3 (Gaschűtz and Itő [10], Satz 5.7, p.436 ). A group G is solvable if all its
minimal subgroups are normal.

2. Proofs

We first prove the following theorems:

Theorem 2. Let P be a Sylow p-subgroup of a group G, where p is an odd prime. If each
subgroup of P of order p is S-permutable in G, then G′is p-nilpotent.

Proof. We prove the theorem by induction on |G| . Hence if each subgroup of P of
order p is normal in G, then each subgroup of G ′ of order p is normal in G′. Let L be a



R. Hijazi, F. Charaf / Eur. J. Pure Appl. Math, 12 (2) (2019), 571-576 573

subgroup of G′ such that |L| = p. Then G/CG(L) ⊆ Aut(L) and, since Aut(L) is cyclic of
order p− 1, we have G/CG(L) is abelian. Thus G′ ≤ CG(L) and so L ≤ Z(G′). By ([10],
Satz 5.5(a), p. 435), G′is p-nilpotent. Thus we may assume that there exists a subgroup H
of P of order p such that H is not normal in G. By the hypothesis, H is S-permutable in
G and hence by ([13], Lemma A), Op(G) ≤ NG(H) < G. Let M be a maximal subgroup
of G such that NG(H) ≤M < G. Then M CG and | G/M |= p. By induction on |G| , M ′
is p-nilpotent. Hence if Op′(G) 6= 1, G/ Op′(G) satisfies the hypothesis of the theorem and
so (G/ Op′(G))′ = G′Op′(G)/Op′(G) ∼= G′/(G8 ∩Op′(G)) is p-nilpotent which implies that
G′is p-nilpotent. Thus assume that Op′(G) = 1. Since M ′ char M and M CG, we have
M ′ C G. As M ′ is p-nilpotent and Op′(G) = 1, we have M ′ is a p-group. Then P1 CM
where P1 is a Sylow p-subgroup of M . By Schur-Zassenhaus Theorem [5, Theorem 6.2.1,
p. 221], M = P1K , where K is a p′-Hall subgroup of M . Hence if CG(P1) ≤ P1, K
is a p′-group of automorphisms of P1, and since K leaves each subgroup of P1 invariant
because every subgroup of P of prime order is S-permutable, then by ([14], Lemma 2.20),
K is cyclic. Let Q be a Sylow q-subgroup of K, where q is a prime divisor of the order of
K. Hence if p < q, then P1Q = P1×Q and this means that Q ≤ CG(P1), a contradiction.
Thus p is the largest prime dividing |G| and since K is cyclic, it follows, by Burnside′s
p-Nilpotent Theorem ([10], Satz 2.8, p.420), that P C G. But G/P ∼= K, therefore G/P
is cyclic and so abelian, then G′ ≤ P . This completes the proof of the theorem.

As a corollary of Theorem 2.1:

Corollary 4. If each subgroup of prime order of G is S-permutable in G, then G is
solvable, S CG′and G′/S is nilpotent,where S is a Sylow 2-subgroup of G′.

Proof. By Theorem 2.1, G′ is p-nilpotent for each odd prime p dividing |G|. So G′ /S
is nilpotent, S is a Sylow 2-subgroup of G′and hence G is solvable.

Theorem 3. Let p be an odd prime and let P be a Sylow p-subgroup of G. Suppose that
P has a subgroup D such that 1 < |D| < |P | and all subgroups H of P with |H| = |D| are
normal in G. Then G′is p-nilpotent.

Proof. We prove the theorem by induction on |G|. Clearly, P∩G′ is a Sylow p-subgroup
of G′. Set P1 = P ∩G′. We deal with the following two cases:

Case 1. |P1| ≤ |D|.
Hence if |D| = p, |P1| = p , and P1 C G. Then G′ ≤ CG(P1) and so P1 ≤ Z(G′).

Hence, by Schur-Zassenhaus Theorem, G′ = P1 ×K, where K is a p′-Hall subgroup of G′.

In particular, G′ is p-nilpotent.
Thus we may assume that |D| = pn (n ≥ 2). Let H be a subgroup of P with |H| = |D|

such that P1 ≤ H < P . By the hypothesis, H CG. Assume that Φ(H) 6= 1 and consider
the factor group G/Φ(H). Obviously, G/Φ(H) satisfies the theorem hypothesis and so
(G/Φ(H))′ = G′Φ(H)/Φ(H) is p-nilpotent by the induction on |G|. But G′Φ(H)/Φ(H) ∼=
G′/G′ ∩ Φ(H) and Φ(H) ≤ Φ(G), then we have G′ ∩ Φ(H) ≤ G′ ∩ Φ(G) and therefore



R. Hijazi, F. Charaf / Eur. J. Pure Appl. Math, 12 (2) (2019), 571-576 574

G′/G′ ∩ Φ(G) is p-nilpotent. Now G′Φ(G)/Φ(G) ∼= G′/G′ ∩ Φ(G) is p-nilpotent implies
that G′Φ(G) is p-nilpotent and consequently G′ is p-nilpotent.

Thus we may assume that Φ(H) = 1 and so H is elementary abelian p-group of order
pn (n ≥ 2). Let L be a subgroup of P contains H such that H is maximal in L. Clearly, L
is not cyclic because H is elementary abelian group of order pn (n ≥ 2). Then L contains a
subgroup H1 such that |H1| = |D| and H1 6= H. By the hypothesis, H1CG and since HCG,
we have L = H1H C G and so Φ(L) ≤ Φ(G). Hence if Φ(L) 6= 1, Φ(L) ≤ H1 < L ≤ P .
Since L is not cyclic, we have Φ(L) is contained properly in H1. Now it is easy to notice
that the factor group G/Φ(L) satisfies the hypothesis of the theorem, so by induction on
|G|, G′ is p-nilpotent. Thus we may assume that Φ(L) = 1 and so P1 is elementary abelian
p-group. Since P1 ≤ H < L ≤ P and H is maximal in L, it follows that |L| = pn+1. Let
L1 =< x1 > be a subgroup of P1 of order p. Then L =< x1 > × < x2 > × . . .× < xn+1 >.
By the hypothesis, each maximal subgroup of L is normal in G. Applying ([1], Lemma
2.9) implies that each subgroup of L of order p is normal in G; in particular each subgroup
L1 of P1 of order p is normal in G. So, G8 ≤ CG(L1) and consequently P1 ≤ Z(G′).
By Schur-Zassenhaus Theorem, G′ = P1 × K1, where K1 is a p′-Hall subgroup of G; in
particular G′ is p-nilpotent.

Case 2. |P1| > |D|.
Hence if |D| = p, then every subgroup of P1 of order p is normal in G, so Ω1(P1) ≤

Z(G′) which implies that G′ is p-nilpotent by ([10], Satz 5.5(a), p 435). Thus assume
that |D| = pn (n ≥ 2). Hence if Φ(D) 6= 1, G/Φ(D) satisfies the hypothesis of the
theorem and so (G/Φ(D))′ = G′Φ(D)/Φ(D) is p-nilpotent by induction on |G| which
implies that G′/G′ ∩ Φ(G) is p-nilpotent; in particular G′ is p-nilpotent. Thus we may
assume that Φ(D) = 1. Let L ≤ P1 such that D is maximal in L. Then |L| = pn+1(n > 2).
Clearly L is not cyclic. Then there exists a maximal subgroup L1 6= D in L. By the
hypothesis L1 C G and D C G which implies that L = L1D C G. Hence if Φ(L) 6= 1,
Φ(L) ≤ D < L ≤ P1 and since L is not cyclic, it follows that Φ(L) < D. By induction
on |G|, G′Φ(L)/Φ(L) ∼= G′/G′ ∩ Φ(L) is p-nilpotent. In particular, G′Φ(G)/Φ(G) is p-
nilpotent and it follows easily that G′ is p-nilpotent. So we may assume that Φ(L) = 1
and so L is elementary abelian. Let L1 < P such that |L1| = p. Then L1 < L ≤ P1

and so L1 CG by ([1], Lemma 2.9). In particular, Ω1(P1) ≤ Z(G′). Again by ([10], Satz
5.5(a), p 435), G′ is p-nilpotent. This completes the proof of the theorem.

Now we can move forward to prove our main theorem:
Proof. We prove the theorem by induction on |G|. Hence if Op′(G) 6= 1, G/Op′(G)

satisfies the hypothesis of the theorem and so (G/Op′(G))′ is p-nilpotent by induction
on |G|; in particular, G′ is p-nilpotent. Thus we may assume that Op′(G) = 1. If each
subgroup H of P with |H| = |D| is normal in G, then G′ is p-nilpotent by Theorem 2.2. So
we may assume that there exists a subgroup H of P with |H| = |D| and H is not normal
in G. By hypothesis, H is S-permutable in G. Since H 6 G and H is S-permutable in G,
we have by ([13], Lemma A) that Op(G) ≤ NG(H) < G. Let M be a maximal subgroup
of G contains NG(H) properly. Then M CG and |G/M | = p. Let P1 = P ∩M be a Sylow
p-subgroup of M . By the hypothesis, |D| ≤ |P1|. If |D| = |P1|, then |H| = |P1| and so



REFERENCES 575

P ≤ NG(H), and since Op(G) ≤ NG(H), we have POp(G) = G ≤ NG(H) < M which is
impossible. Thus we may assume that |D| < |P1|. Now M ′ is p-nilpotent, by the inductive
hypothesis, implies that M ′ is a p-group because Op′(G) = 1. Then P1 is characteristic in
M and since MCG, we have P1CG. If PCG, then G/P is abelian and since all subgroups
H of P with |H| = |D| are S-permutable in G, we have that G is supersolvable by ([14],
Theorem 1.3) and so G′ is nilpotent; in particular G′ is p-nilpotent. Thus we may assume
that P 6 G and P1 = F (G) the Fitting subgroup of G (recall that Op′(G) = 1 and that
F (G) =< Op(G) for all p divides |G| >). Consider the subgroup Φ(P1) and assume that
Φ(P1) 6= 1. Hence if |Φ(P1)| < |D|, then (G/Φ(P1))

′ is p-nilpotent by induction on |G|; in
particular G′ is p-nilpotent. So assume that |Φ(P1)| ≥ |D|. If |Φ(P1)| = |D|, then P/Φ(P1)
is not cyclic. Let L/Φ(P1) be a proper subgroup of P/Φ(P1) such that |L/Φ(P1)| = p (L
is not cyclic; otherwise Φ(P1) is cyclic and this implies that there exists L1 ≤ Φ(P1) such
that L1CG; in particular G/CG(L1) is isomorphic to a subgroup of Aut(L1) and so G′ ≤
CG(L1) and we conclude then that G′ is p-nilpotent). As |L/Φ(P1)| = p, then there exists
a maximal subgroup L1 of L such that |L1| = |Φ(P1)| = |D| and L1 6= Φ(P1). But L1Φ(P1)
is S-permutable in G, then L1Φ(P1)/Φ(P1) = L/Φ(P1) is S-permutable in G/Φ(P1). By
Theorem 2.1, (G/Φ(P1))

′ = G′Φ(P1)/Φ(P1) is p-nilpotent and so G′ is p-nilpotent. Thus
we may assume that Φ(P1) = 1 and P1 is elementary abelian. Since all subgroups H of P1

with |H| = |D| are normal in M , we have by ([1], Lemma 2.9) that all subgroups of P1 of
order p are normal in M . So P1 ∩ Z(P ) 6= 1. Let L ≤ P1 ∩ Z(P ) such that |L| = p. Then
L C G and since G/CG(L) is isomorphic to a subgroup of Aut(L), we have that G′ ≤
CG(L), in particular G′L/L is p-nilpotent and so G′ is p-nilpotent. This completes the
proof of the theorem.

References

[1] M. Asaad and A. A. Heliel, On S-quasinormally embedded subgroups of finite groups,
JPAA 165(2001) 129-135.

[2] A. Ballester-Bolinches and X. Guo, On complemented subgroups of finite groups,
Arch. Math. 72(1999) 161-166.

[3] A. Ballester-Bolinches, Y. Wang and X. Guo (2000), C-supplemented subgroups of
finite groups, Glasgow Math. J. 42(2000) 383-389.

[4] J. Buckely, Finite groups whose minimal subgroups are normal, Math. Z. 116(1970)
15-17.

[5] D. Gorenstein, Finite Groups, American Mathematical Society, 1980.

[6] P. Hall, A characteristic property of solvable groups, J. London Math. Soc. 12(1937)
198-200.

[7] P. Hall, Complemented Groups, J. London Math. Soc. 12(1937) 201-204.



REFERENCES 576

[8] A. A. Heliel, A note on c-supplemented subgroups of finite groups, Comm. Algebra
42(2014) 1650-1656.

[9] R. Hijazi, A note on solvability of finite groups, Journal of Advances in Mathematics
10(2015) 3639-3642.

[10] B. Huppert, Endliche Gruppen I, Springer, Berlin-New York, 1979.

[11] O. H. Kegel, Sylow-Gruppen und Subnormalteiler endlicher Gruppen, Math. Z.
78(1962) 205-221.

[12] J. Li, W. Shi, G. Chen and D. Yu, New characterization of solubility of finite groups,
Italian J. Pure Appl. Math. 33(2014) 377-382.

[13] P. Schmid, Subgroups permutable with all Sylow subgroups, J. Algebra, 207 (1998)
285-293.

[14] A. N. Skiba, On weakly S-permutable subgroups of finite groups, J. Algebra,
315(2007) 192-209.

[15] S. Srinivasan, Two sufficient conditions for supersolvability of finite groups, Israel J.
Math. 35(1980) 210-214.