Finite group with given c-permutable subgroups
Following [1] we say that subgroups H and T of a group G are c-permutable in G if there exists an element x ∈ G such that HTˣ = TˣH. We prove that a finite soluble group G is supersoluble if and only if every maximal subgroup of every Sylow subgroup of G is c-permutable with all Hall subgro...
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Інститут прикладної математики і механіки НАН України
2004
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| Цитувати: | Finite group with given c-permutable subgroups / Ahmad Alsheik Ahmad // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 2. — С. 9–16. — Бібліогр.: 12 назв. — англ. |
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irk-123456789-1564632019-06-19T01:27:35Z Finite group with given c-permutable subgroups Ahmad Alsheik Ahmad Following [1] we say that subgroups H and T of a group G are c-permutable in G if there exists an element x ∈ G such that HTˣ = TˣH. We prove that a finite soluble group G is supersoluble if and only if every maximal subgroup of every Sylow subgroup of G is c-permutable with all Hall subgroups of G. 2004 Article Finite group with given c-permutable subgroups / Ahmad Alsheik Ahmad // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 2. — С. 9–16. — Бібліогр.: 12 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20D10. http://dspace.nbuv.gov.ua/handle/123456789/156463 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Following [1] we say that subgroups H and T of
a group G are c-permutable in G if there exists an element x ∈ G
such that HTˣ = TˣH. We prove that a finite soluble group G is
supersoluble if and only if every maximal subgroup of every Sylow
subgroup of G is c-permutable with all Hall subgroups of G. |
| format |
Article |
| author |
Ahmad Alsheik Ahmad |
| spellingShingle |
Ahmad Alsheik Ahmad Finite group with given c-permutable subgroups Algebra and Discrete Mathematics |
| author_facet |
Ahmad Alsheik Ahmad |
| author_sort |
Ahmad Alsheik Ahmad |
| title |
Finite group with given c-permutable subgroups |
| title_short |
Finite group with given c-permutable subgroups |
| title_full |
Finite group with given c-permutable subgroups |
| title_fullStr |
Finite group with given c-permutable subgroups |
| title_full_unstemmed |
Finite group with given c-permutable subgroups |
| title_sort |
finite group with given c-permutable subgroups |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2004 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/156463 |
| citation_txt |
Finite group with given c-permutable subgroups / Ahmad Alsheik Ahmad // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 2. — С. 9–16. — Бібліогр.: 12 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT ahmadalsheikahmad finitegroupwithgivencpermutablesubgroups |
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2025-07-14T08:50:01Z |
| last_indexed |
2025-07-14T08:50:01Z |
| _version_ |
1837611615662374912 |
| fulltext |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 2. (2004). pp. 9 – 16
c© Journal “Algebra and Discrete Mathematics”
Finite group with given c-permutable subgroups
Ahmad Alsheik Ahmad
Communicated by L. A. Shemetkov
Abstract. Following [1] we say that subgroups H and T of
a group G are c-permutable in G if there exists an element x ∈ G
such that HT x = T xH. We prove that a finite soluble group G is
supersoluble if and only if every maximal subgroup of every Sylow
subgroup of G is c-permutable with all Hall subgroups of G.
Introduction
All considered in this paper groups are finite. It is interest to use some
information on Sylow subgroups of a group G to determine the structure
of the group. For instance, the knowledge of the maximal subgroups of
Sylow subgroups often yields wealth of information about the group itself.
In [2] Srinivasan proved that G is supersoluble if every maximal subgroup
of every Sylow subgroup of G is normal in G. In [3] Wang introduced the
concept of c-normality which is a weaker condition than the normality in
deriving the same result. In [4,5] the supersolubility of groups in which
all maximal subgroups of all Sylow subgroups are complemented was
proved. In the paper [6] (see also [7]) Guo, Shum and Skiba proved that
a group G is supersoluble if and only if every non-normal in G maximal
subgroup of every Sylow subgroup of G has a supersoluble supplement
in G. The analogous results for p-nilpetent and p-closed groups were
obtained by Kosenok in [8]. In this paper we give in this direction two
new characterizations of supersoluble groups appealing to the following
concept of c-permutability which was introduced in [1].
2000 Mathematics Subject Classification: 20D10.
Key words and phrases: finite group, maximal subgroup, Sylow subgroup, su-
persoluble group, c-permutable subgroup.
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.10 Finite group with given c-permutable subgroups
Let H and T be subgroups of a group G. Then, H and T are said to
be conditionally permutable (or in brevity, c-permutable) in G if HT x =
T xH, for some x ∈ G.
The condition of c-permutability is generally weaker than the condi-
tion of permutability, for example, one can see that a Sylow 2-subgroup
of the simmetric group S3 is not permutable, but it is c-permutable with
all subgroups of S3.
1. Preliminaries
All considered later on groups are soluble. In this section we give some
known results about soluble and supersoluble groups which will be needed
in proving our main results later on.
The following lemma is well known.
Lemma 1. Let H be a Hall π-subgroup of a group G and K E G. Then
the following statements hold:
(1) HK/K is a Hall π-subgroup of G/K;
(2) H ∩ K is a Hall π-subgroup of K;
(3) If G = AB for some subgroups A, B of G, then there exist Hall
π-subgroup Gπ, Aπ and Bπ in G, A and B respectively such that
Gπ = AπBπ.
Lemma 2. Let H be a subnormal subgroup of a group G. Then:
(1) If H ≤ K ≤ G, then H is subnormal in K;
(2) If H is a π-group, then H ≤ Oπ(G) (see [9; Corollary 7.7.2).
Now we cite some properties of the supersoluble groups from the
literature.
Lemma 3. Let G be a group. Then the following statements hold:
(i) If G is supersoluble, then G′ ⊆ F (G) and G is p-closed for the
largest prime divisor p of |G| (see [10; VI, 9.11]);
(ii) If L E G and L/Φ(G) ∩ L is supersoluble, then L is supersoluble
(see [9; Corollary 4.2.4]);
(iii) G is supersoluble if and only if |G : M | is a prime for every maximal
subgroup M of G (see [10; VI, 9.3]).
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.A. Alsheik Ahmad 11
(vi) G is supersoluble if it has two normal subgroups A and B with
supersoluble quotients such that A ∩ B = 1.
We use the symbol F (G) to denote the Fitting subgroup of a group
G (that is the product of all normal nilpotent subgroups of G).
Lemma 4 ([11]; A,(10.6)). The Fitting subgroup F (G) of a group G
has the following properties:
(1) F (G) ≤ CG(H/K) for every chief factor H/K of G;
(2) CG(F (G)) ⊆ F (G).
Lemma 5. If L E G and L/Φ(G) ∩ L is nilpetent, then L is nilpotent
(see [9; Corollary 4.2.1]);
Lemma 6 ([1], Lemma 2.1). Let G be a group. Suppose that K E G
and H ≤ G. Then:
(i) If K ≤ T ≤ G and H is c-permutable with T in G, then KH/K is
c-permutable with T/K in G/K;
(ii) If K ≤ H, T ≤ G and H/K is c-permutable with KT/K in G/K,
then H is c-permutable with T in G.
Lemma 7. ([12], Lemma 1.5.6). Let H/K be an abelian chief factor of
a group G. Let M be a maximal subgroup of G such that K ⊆ M and
MH = G. Then
G/MG ' [H/K](G/CG(H/K)).
2. Main results
Theorem 1. A soluble group G is supersoluble if and only if it has a non-
identity normal subgroup N with supersoluble quotient such that every
maximal subgroup of every Sylow subgroup of N is c-permutable with all
Hall subgroups of G.
Proof. First suppose that G has a non-identity normal subgroup N with
supersoluble quotient such that every maximal subgroup of every Sylow
subgroup of N is c-permutable with all Hall subgroups of G. We shall
show that G is supersoluble. Assume that it is false and let G be a
counterexample of minimal order. Then the following statements hold.
(a) G/K is supersoluble for every non-identity normal in G sub-
group K.
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.12 Finite group with given c-permutable subgroups
By hypothesis, it is true if K = N . Let K 6= N . We shall show that
the hypothesis of the theorem is true for G/K. First of all we note that
KN/N is such a non-identity normal in G/K subgroup that the quotient
(G/K)(KN/K) ' G/KN ' (G/N)/(KN/N)
is supersoluble.
Let P/K be a Sylow p-subgroup of KN/K and P1/K be a maximal
in P/K subgroup. Let E/K be a Hall π-subgroup of KN/K. We have to
prove that P1/K is c-permutable with E/K. Let R be a Sylow p-subgroup
of KN such that RK/K = P/K. By Lemma 1 there exist Sylow p-
subgroups Np and Kp in N and K respectively such that R = NpKp.
Hence P/K = NpK/K. We will show that P1∩Np is a maximal subgroup
of Np. First of all we note that P1 ∩ Np 6= Np. Indeed, if P1 ∩ Np = Np,
then Np ⊆ P1 and so
P/K = NpK/K = P1/K,
this contradicts the choice of the subgroup P1/K. Next assume that G
has a subgroup T such that P1 ∩ Np ⊂ T ⊂ Np. Then
P1 = K(P1 ∩ Np) ⊆ TK ⊆ KNp = P.
But P1 is a maximal subgroup of P and so we have either P1 = TK or
TK = KNp. If P1 = TK, we have T ⊆ P1 ∩ Np ⊆ T , that is impossible.
Hence TK = NpK and therefore
Np = Np ∩ TK = T (Np ∩ K) ⊆ T (P1 ∩ Np) = T,
a contradiction. Hence we have to conclude that P1 ∩ Np is a maximal
subgroup of Np. By hypothesis it follows that P1 ∩ Np is c-permutable
with all Hall subgroups of G. By Lemma 1, G has a Hall π-subgroup
Gπ such that E/K = GπK/K. Let x be an element of G such that
Gx
π(P1 ∩ Np) = (P1 ∩ Np)G
x
π. Then
Gx
π(P1 ∩ Np)K/K = Gx
πP1/K = (Gx
πK/K)(P1/K) =
= (E/K)xK(P1/K) = (P1/K)(E/K)xK = (P1 ∩ Np)G
x
πK/K.
(b) If H is a minimal normal subgroup of G, then for some prime p
we have
H = Op(G) = F (G) = CG(H) * Φ(G)
and |H| 6= p.
By Lemma 3 and Statement (a), H is the unique minimal normal
subgroup of G and H * Φ(G). Let M be a maximal subgroup of G such
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.A. Alsheik Ahmad 13
that H * M . Since G is soluble, H is an elementary abelian p-group for
some prime p. Hence G = [H]M . Now let C = CG(H). Then
C = C ∩ HM = H(C ∩ M).
But, evidently, C ∩ M E G and so C ∩ M = 1. It follows that H = C.
Besides, since by Lemma 4 we have F (G) ⊆ C, then H = F (G) = Op(G).
It is also clear that |H| 6= p.
(c) H is not a Sylow subgroup of N .
Suppose that H is a Sylow p-subgroup of N . Let M be a maximal
subgroup of H, Q be a Sylow q-subgroup of G where q 6= p. By hypothesis
for some x ∈ G we have D = MQx = QxM . Since M is subnormal in
G, then by Lemma 2, M is subnormal in D too. But M is a Sylow
subgroup of D, and so by Lemma 2, M E D. Hence q - |G : NG(M)|.
Thus |G : |G : NG(M)| = pα for some α ∈ {0} ∪ N. But in view
of the minimality of H and by Statement (b), NG(M) 6= G. Hence
p | |G : NG(M)| and so p | n where n is the number of all maximal in H
subgroups. This contradicts Statement 8.5 from [10; III]. So we have (c).
(d) N = G.
Indeed, suppose that N 6= G. Let Nq be a Sylow subgroup of N , P1
be a maximal subgroup of Nq. And let T be a Hall π-subgroup of N .
Let us choose in G a Hall π-subgroup Gπ such that T ⊆ Gπ. Then
T = N ∩ Gπ. By hypothesis G has an element x such that D = P1G
x
π =
Gx
πP1. It follows that
N ∩ D = P1(N ∩ Gx
π) = (N ∩ Gx
π)P1.
But by Lemma 1, N∩Gx
π is a Hall π-subgroups of N . Thus every maximal
subgroup of every Sylow subgroup of N is c-permutable with all Hall
subgroup of N . Thus the hypothesis of this theorem is true for N . But
|N | < |G| and so by the choice of G we have to conclude that N is
supersoluble. Let q be the largest prime divisor of |N |. Then by Lemma 3,
a Sylow q-subgroup Nq of N is normal in N . Since Nq char NEG it follows
that Nq E G. Then Nq ⊆ F (G) and hence by (b) we have Nq ⊆ H. Thus
Nq = H, this contradicts Statement (c). This contradiction completes
the proof of Statements (d).
(e) Conclusion contradiction.
Let P be a Sylow p-subgroup of G. Then H ⊆ P and by State-
ment (c), H 6= P . If H ≤ Φ(P ), then by Statement 3.2 from [10;
III], H ⊆ Φ(G), contrary to Statement (b). Hence one can choose in
P a maximal subgroup P1 such that H * P1. Let D be a Hall p′-
subgroup of G. Then by hypothesis there exists an element x such that
T = P1D
x = DxP1. It is clear that |G : T | = |P : P1| = p. Besides,
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.14 Finite group with given c-permutable subgroups
evidently, H * T and so 1 6= H ∩ T 6= H. But H ∩ T E G, this con-
tradicts the minimality of H. This contradiction completes the proof of
supersolubility G.
Finally suppose that G is a supersoluble group. Let N be a minimal
normal subgroup of G. Then |N | = p for some prime p. Thus a maximal
subgroup of N is normal in G.
Theorem 2. A soluble group G is supersoluble if and only if it has a non-
identity normal subgroup N with supersoluble quotient such that every
non-complemented in G maximal subgroup of every Sylow subgroup of
F (N) is c-permutable with all subgroups of G.
Proof. We need only prove the “if” part (see the proof of Theorem 1).
Assume that this is false and let G be a counterexample of minimal
order.
Let Φ be a minimal normal subgroup of G contained in N . Sup-
pose that Φ ⊆ Φ(G). Then (since by hypothesis G/N is supersolu-
ble) in view of Lemma 3, Φ 6= N . Consider the quitient G/Φ. Let
T/Φ = F (N/Φ). Since T/Φ is nilpotent, then by Lemma 5, T is a nilpo-
tent normal subgroup of N . Hence T ⊆ F (N). On the other hand, since
F (N)/Φ ⊆ F (N/Φ), we have F (N) ≤ T and so T = F (N). Thus
F (N/Φ) = T/Φ = F (N)/Φ.
Now let P/Φ be a Sylow p-subgroup of T/Φ, M/Φ be a maximal in
P/Φ subgroup and Pp be a Sylow p-subgroup of P . Then PpΦ = P and
L = M ∩ Pp is a maximal subgroup of Pp (see the proof of Theorem 1).
Hence by hypothesis L is either complemented in G or c-permutable with
all subgroups of G. Assume we have the first case and let T be a subgroup
of G such that G = LT and L∩T = 1. We shall show that then TΦ/Φ is
a complement to M/Φ in G/Φ. It is clear that G/Φ = (LΦ/Φ)(TΦ/Φ).
Suppose that Φ is a q-group where q 6= p. Then, evidently, Φ ⊆ T and so
(LΦ/Φ) ∩ (T/Φ) = (LΦ ∩ T )/Φ = Φ(L ∩ T )/Φ = Φ/Φ
. Analogously one can show that TΦ/Φ is a complement to M/Φ in G/Φ
in the case when Φ is a p-group. Finally, if we have the second case
then by Lemma 6 we see that M/Φ = LΦ/Φ is c-permutable with all
subgroups of G/Φ. Therefore G/Φ has a non-identity normal subgroup
N/Φ with supersoluble quotient
(G/Φ)/(N/Φ) ' G/N
such that every non-complemented in G/Φ maximal subgroup of every
Sylow subgroup of F (N/Φ) = F (N)/Φ is c-permutable with all subgroups
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.A. Alsheik Ahmad 15
of G/Φ. Thus the hypothesis is true for G/Φ. Since Φ 6= 1, |G/Φ| < |G|
and so by the choice of G, G/Φ is supersoluble. But then by Lemma 3, G
is supersoluble too, a contradiction. Hence Φ * Φ(G). Thus Φ(G)∩N =
1. Since N EG and F (N) char N , F (N)EG. It follows Φ(F (N)) ⊆ Φ(G)
and so Φ(F (N)) = 1. Thus F (N) is an abelian group.
Now let D be the least (by inclusion) normal subgroup of G with
supersoluble quotient. Then D ≤ N and so if R is a minimal normal
subgroup of G contained in D, then R ⊆ F (N). Since Φ(G) ∩ N = 1, G
has a maximal subgroup M such that G = [R]M and so |G : M | = |R|.
First assume that |R| is a prime. Then by Lemma 7,
G/MG ' [RMG/MG]CG(RMG/MG)
is a metacyclic group. Hence G/MG is a supersoluble group and so
R ⊆ D ⊆ MG ⊆ M.
But then G = RM = M . This contradiction shows that |R| is not a
prime.
Let R1 be a maximal subgroup of R. Then R1 6= 1. Let R be a
p-group and D = Op(N). Let us choose a maximal subgroup E of G
such that G = ER. Then G = DE and so K = E ∩ D E G. Since,
evidently, E ∩R = 1, D = R×K. Let M = R1K. Then since |D : M | =
|R : R1| = p, M is a maximal subgroup of D. Hence by hypothesis M is
either c-permutable with all subgroups of G or it is complemented in G.
If we have the first case, then, in particular,
R1KEx = R1E
x = ExM
for some x ∈ G. It follows that |G : E| = |R1| = |R|. This contradiction
shows that we have the second case. Let T be a complement to M in G.
It is not difficult to show that V = T ∩D is a group of prime order p. It
is clear also that V E G. Because D/K is a chief factor of G and since
KV is a normal in G subgroup such that K ⊆ KV ⊆ D, it follows that
KV = D and so p = |V | = |D/K| = |R|. This contradiction completes
the proof of this theorem.
References
[1] W. Guo, K.P.Shum and A. N. Skiba, Conditionally Permutable Subgroups and
Supersolubility of Finite Groups, Preprint No 49 , Gomel State University, 2003.
[2] S.Srinivasan, Two sufficient conditions for supersolubility of finite groups, Israel
J. Math., 35, 1990, pp.210-214.
[3] Y. Wang, c-Normality of Groups and its Properties, J.Algebra, 78, 1996, pp.101-
108.
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.16 Finite group with given c-permutable subgroups
[4] A. Ballester-Bolinches, H. Guo, On complemented subgroups of finite groups,
Arch.Math.(Basel), 72, 1999, pp.161-166.
[5] Y. Wang, Finite groups with some subgroups of Sylow subgroups c- supplemented
, J.Algebra, 224, 224, pp.467- 478.
[6] W. Guo, K.P.Shum and A. N. Skiba, G-Covering Subgroup Systems for the Classes
of Supersoluble and Nilpotent Groups, Israel J. Math.,138, 2003, pp. 125-138.
[7] W. Guo, K.P.Shum and A. N. Skiba, G-Covering Subgroup Systems for the Classes
of Supersoluble and Nilpotent Groups, Preprint No 21, Gomel State University,
2001.
[8] N. Kosenok, Criterions of p-closure and p-nilpotency of finite groups, Vesty NAN
Belarusy, Ser. fiz.-mat. nauk, 4, 2003, pp.68-73.
[9] L.A. Shemetkov, Formations of Finite Groups, Nauka, Moscow, 1978.
[10] B. Huppert, Endliche Gruppen I, Springer–Verlag, Berlin–Heidelberg–New York,
1979.
[11] K. Doerk, O. Hawkes, Finite Soluble Groups, Walter de Gruyter, Berlin-New
York, 1992.
[12] W. Guo, The Theory of Classes of Groups, Science Press /Kluwer Academic
Publishers, Beijing–New York–Dordrecht–Boston–London, 2000.
Contact information
A. Alsheik Ahmad Belorussian State University of Transport,
Belarus, 246017, Gomel, Krasnaarmeyskaya
Str. 4a, 403
E-Mail: belgut@tut.by
Received by the editors: 17.05.2004.
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