Characterization of finite groups with some S-quasinormal subgroups of fixed order
Let G be a finite group. A subgroup of G is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G. We fix in every non-cyclic Sylow subgroup P of the generalized Fitting subgroup a subgroup D such that 1 < |D| < |P| and characterize G under the assumption that all subgrou...
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2012
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| Цитувати: | Characterization of finite groups with some S-quasinormal subgroups of fixed order / M. Asaad, P. Csorgo // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 161–167. — Бібліогр.: 11 назв. — англ. |
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irk-123456789-1522292019-06-10T01:26:11Z Characterization of finite groups with some S-quasinormal subgroups of fixed order Asaad, M. Let G be a finite group. A subgroup of G is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G. We fix in every non-cyclic Sylow subgroup P of the generalized Fitting subgroup a subgroup D such that 1 < |D| < |P| and characterize G under the assumption that all subgroups H of P with |H| = |D| are S-quasinormal in G. Some recent results are generalized. 2012 Article Characterization of finite groups with some S-quasinormal subgroups of fixed order / M. Asaad, P. Csorgo // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 161–167. — Бібліогр.: 11 назв. — англ. 1726-3255 2000 MSC:20D10, 20D30. http://dspace.nbuv.gov.ua/handle/123456789/152229 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let G be a finite group. A subgroup of G is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G. We fix in every non-cyclic Sylow subgroup P of the generalized Fitting subgroup a subgroup D such that 1 < |D| < |P| and characterize G under the assumption that all subgroups H of P with |H| = |D| are S-quasinormal in G. Some recent results are generalized. |
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Article |
| author |
Asaad, M. |
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Asaad, M. Characterization of finite groups with some S-quasinormal subgroups of fixed order Algebra and Discrete Mathematics |
| author_facet |
Asaad, M. |
| author_sort |
Asaad, M. |
| title |
Characterization of finite groups with some S-quasinormal subgroups of fixed order |
| title_short |
Characterization of finite groups with some S-quasinormal subgroups of fixed order |
| title_full |
Characterization of finite groups with some S-quasinormal subgroups of fixed order |
| title_fullStr |
Characterization of finite groups with some S-quasinormal subgroups of fixed order |
| title_full_unstemmed |
Characterization of finite groups with some S-quasinormal subgroups of fixed order |
| title_sort |
characterization of finite groups with some s-quasinormal subgroups of fixed order |
| publisher |
Інститут прикладної математики і механіки НАН України |
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2012 |
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http://dspace.nbuv.gov.ua/handle/123456789/152229 |
| citation_txt |
Characterization of finite groups with some S-quasinormal subgroups of fixed order / M. Asaad, P. Csorgo // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 161–167. — Бібліогр.: 11 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
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AT asaadm characterizationoffinitegroupswithsomesquasinormalsubgroupsoffixedorder |
| first_indexed |
2025-07-13T02:36:03Z |
| last_indexed |
2025-07-13T02:36:03Z |
| _version_ |
1837497494442868736 |
| fulltext |
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h.Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 14 (2012). Number 2. pp. 161 – 167
c© Journal “Algebra and Discrete Mathematics”
Characterization of finite groups
with some S-quasinormal subgroups
of fixed order
M. Asaad and Piroska Csörgő
Communicated by V. I. Sushchansky
Abstract. Let G be a finite group. A subgroup of G is
said to be S-quasinormal in G if it permutes with every Sylow
subgroup of G. We fix in every non-cyclic Sylow subgroup P of the
generalized Fitting subgroup a subgroup D such that 1 < |D| < |P |
and characterize G under the assumption that all subgroups H of
P with |H| = |D| are S-quasinormal in G. Some recent results are
generalized.
1. Introduction
All groups considered in this paper are finite. The terminology and
notations employed agree with standard usage, as in Huppert [5]. Two
subgroups H and K of a group G are said to permute if KH = HK. It is
easily seen that H and K permute if and only if the set HK is a subgroup
of G. We say, following Kegel [7], that a subgroup of G is S-quasinormal
in G, if it permutes with every Sylow subgroup of G.
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
This paper was partly supported by Hungarian National Foundation for Scientific
Research Grant # K84233.
2000 MSC: 20D10, 20D30.
Key words and phrases: S-quasinormality, generalized Fitting subgroup, super-
solvability.
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162 Characterization of finite groups
chief factor of G. Clearly F ∗(G) is a characteristic subgroup of G and
F ∗(G) 6= 1 if G 6= 1 (see in [6, X. 13]). By [5, III. 4.3] F (G) 6 F ∗(G).
A number of authors have examined the structure of a finite group
G under the assumption that all subgroups of G of prime order are well-
situated in G. The authors [1] showed that if G is a solvable group and
every subgroup of F (G) of prime order or of order 4 is S-quasinormal in G,
then G is supersolvable. Li and Wang [8] showed that if G is a group and
every subgroup of F ∗(G) of prime order or of order 4 is S-quasinormal in
G, then G is supersolvable.
Yao, Wang and Li in [4] gave a revised version of our earlier result
in [2]:
Theorem 1.1 ([4, Theorem 1’]). Let G be a group of composite order
such that G is quaternion-free. Suppose G has a nontrivial normal sub-
group N such that G/N is supersolvable. Then the following statements
are equivalent:
(1) Every subgroup of F ∗(N) of prime order is S-quasinormal in G.
(2) G = UW , where U is a normal nilpotent Hall subgroup of odd order,
W is a supersolvable Hall subgroup with (|U |, |W |) = 1 and every
subgroup of F (N) of prime order is S-quasinormal in G.
(3) N is solvable and every subgroup of F (N) of prime order is S-quasi-
normal in G.
In this paper we generalize this theorem: instead of requiring the
S-quasinormality of every subgroup of F ∗(N) of prime order we fix in
every non-cyclic Sylow subgroup P of F ∗(N) a subgroup D such that
1 < |D| < |P | and characterize G under the assumption that all subgroups
H of P with |H| = |D| are S-quasinormal in G.
Theorem 1.2. Let G be a group of composite order such that G is
quaternion-free. Suppose that G has a nontrivial normal subgroup N such
that G/N is supersolvable. Then the following statements are equivalent:
(1) Every non-cyclic Sylow subgroup P of F ∗(N) has a subgroup D
such that 1 < |D| < |P | and all subgroups H of P with |H| = |D|
are S-quasinormal in G.
(2) G = UW , where U is a normal nilpotent Hall subgroup of G of odd
order, W is a supersolvable Hall subgroup of G with (|U |, |W |) = 1,
every non-cyclic Sylow subgroup P of F (N) of odd order has a
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M. Asaad, P. Csörgő 163
subgroup D such that 1 < |D| < |P | and all subgroups H of P with
|H| = |D| permute with R, where R is any Sylow subgroup of G
with (|R|, |U |)=1 and O2(N) 6 Z∞(G).
(3) N is solvable and every non-cyclic Sylow subgroup P of F (N) has
a subgroup D such that 1 < |D| < |P | and all subgroups H of P
with |H| = |D| are S-quasinormal in G.
2. Preliminaries
Lemma 2.1 ([2, Lemma 2.1]). Suppose that G is a quaternion-free group
and every subgroup of G of order 2 is normal in G. Then G is 2-nilpotent.
Lemma 2.2 ([4, Lemma 2]). Suppose that G is a quaternion-free group.
If every subgroup of G of order 2 is S-quasinormal in G, then G is
2-nilpotent.
Lemma 2.3 ([3]). Let p be the smallest prime dividing |G| and let P be a
Sylow p-subgroup of G. If every maximal subgroup of P is S-quasinormal
in G, then G is p-nilpotent.
Lemma 2.4 ([7]). Let G be a group and H 6 K 6 G. Then
(1) If H is S-quasinormal in G, then H is S-quasinormal in K.
(2) Suppose that H is normal in G. Then K/H is S-quasinormal in G/H
if and only if K is S-quasinormal in G.
Lemma 2.5 ([9]). Let G be a group and let P be an S-quasinormal
p-subgroup of G, where p is a prime. Then Op(G) 6 NG(P ).
As an immediate consequence of [10, Theorem 1.3], we have
Lemma 2.6. Let G be a group with a normal subgroup N such that
G/N is supersolvable. Suppose that every non-cyclic Sylow subgroup P of
F ∗(N) has a subgroup D such that 1 < |D| < |P | and all subgroups H of
P with order |H| = |D| and with order 2|D| (if P is a non-abelian 2-group
and |P : D| > 2) are S-quasinormal in G. Then G is supersolvable.
3. Main results
As an improvement of Lemma 2.1, we have
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164 Characterization of finite groups
Lemma 3.1. Suppose that G is a quaternion-free group. Let P be a
Sylow 2-subgroup of G and D 6 P with 2 6 |D| < |P |. If every subgroup
of P of order |D| is normal in G, then G is 2-nilpotent.
Proof. Suppose that the lemma is false and let G be a counterexample of
minimal order. If |D| = 2, then every subgroup of P of order 2 is normal
in G and hence G is 2-nilpotent by Lemma 2.1, a contradiction. Thus
we may assume that 2 < |D| < |P |. Let H be a subgroup of P such that
|H| = |D|. Then H is normal in G by the hypothesis of the lemma. Let
K be a subgroup of P such that H 6 K and |K| = 2|H|. It is clear that
HR is a subgroup of G, where R is any Sylow subgroup of G of odd
order. If H is cyclic, then R is normal in HR by [5]. If H is not cyclic,
then K is not cyclic. Hence there exists a maximal subgroup L of K such
that L 6= H. Clearly, K = HL. By the hypothesis of the lemma H and
L are normal in G. Then K is normal in G, so KR is a subgroup of G.
Clearly, all maximal subgroups of K are normal in KR. Then R is normal
in KR by Lemma 2.3 and so R is normal in HR. Thus HR = H × R,
where R is any Sylow subgroup of G of odd order. Then by [11, p. 221],
1 6= H 6 Z∞(G), so Z(G) 6= 1. Let A 6 Z(G) such that |A| = 2. Then
A is normal in G. Now consider G/A. Clearly, every subgroup of P/A of
order |D|
|A| is normal in G/A (recall that |D| > 2). Then G/A is 2-nilpotent
by our minimal choice of G and since A 6 Z(G), it follows that G is
2-nilpotent, a contradiction.
As an improvement of Lemma 2.2, we have
Lemma 3.2. Suppose that G is a quaternion-free group. Let P be a
Sylow 2-subgroup of G and D 6 P with 2 6 |D| < |P |. If every subgroup
of P of order |D| is S-quasinormal in G, then G is 2-nilpotent.
Proof. Suppose that the lemma is false and let G be a counterexample of
minimal order. Then there exists a subgroup H of P of order |D| such
that H is not normal in G by Lemma 3.1. By the hypothesis, H is S-
quasinormal in G. Then by Lemma 2.5 O2(G) 6 NG(H) < G. Let M be
a maximal subgroup of G such that NG(H) 6 M < G. Then |G/M | = 2.
Let M2 be a Sylow 2-subgroup of M . If |D| = |M2|, then every maximal
subgroup of P is S-quasinormal in G and so G is 2-nilpotent by Lemma 2.3,
a contradiction. Thus we may assume that M2 has a subgroup D such
that 2 6 |D| < |M2|. By Lemma 2.4 (1), every subgroup of M2 of order
|D| is S-quasinormal in M . Then M is 2-nilpotent by the minimal choice
of G and so G is 2-nilpotent, a contradiction.
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M. Asaad, P. Csörgő 165
As an immediate consequence of Lemma 3.2, we have
Lemma 3.3. Suppose that G is a quaternion-free group. Let P be a Sylow
2-subgroup of G. If P is cyclic or P has a subgroup D with 2 6 |D| < |P |
such that every subgroup of P of order |D| is S-quasinormal in G, then
G is 2-nilpotent.
As a corollary of the proofs of Lemmas 3.1 and 3.2, we have
Lemma 3.4. Let G be a group of odd order, p be the smallest prime
dividing |G| and P a Sylow p-subgroup of G. If P is cyclic or P has a
subgroup D with p 6 |D| < |P | such that every subgroup of P of order
|D| is S-quasinormal in G, then G is p-nilpotent.
Lemma 3.5. Let G be a supersolvable group of composite order. Then
G = UW , where U is a normal nilpotent Hall subgroup of G of odd order,
W is a supersolvable Hall subgroup of G with (|U |, |W |) = 1.
Proof. Since G is supersolvable of composite order, it follows that G
possesses a Sylow tower of supersolvable type. Hence P is normal in G,
where P is a Sylow p-subgroup of G and p (p > 2) is the largest prime
dividing |G|. Let U be a normal nilpotent Hall subgroup of G of odd
order such that P 6 U . By the Schur–Zassenhaus Theorem, G has a
Hall subgroup W such that G = UW with (|U |, |W |) = 1. Clearly W is
supersolvable.
Proof of Theorem 1.2. By the hypothesis of the theorem N is a nontrivial
subgroup of G. Then F ∗(N) 6= 1 (see [6, X. 13]).
(1) =⇒ (2) If every noncyclic Sylow subgroup P of F ∗(N) has a
subgroup D such that 1 < |D| < |P | and all subgroups H of P with
|H| = |D| are S-quasinormal in G, then all subgroups H of P with
|H| = |D| are S-quasinormal in F ∗(N) by Lemma 2.4 (1). By Lemmas 3.3
and 3.4, F ∗(N) possesses an ordered Sylow tower of supersolvable type.
Then F ∗(N) is solvable and so F ∗(N) = F (N) (see [6, Ch. X. 13]).
Let p be the smallest prime dividing |F (N)|. If p = 2, then O2(N) 6= 1
and O2(N)R is a subgroup of G for any Sylow subgroup R of G of odd
order. Then by Lemmas 2.4 (1) and 3.3, O2(N)R is 2-nilpotent. Hence
O2(N)R = O2(N)×R for any Sylow subgroup R of G of odd order. Now it
follows easily that every subgroup of O2(N) of order 2|D| is S-quasinormal
in G. Hence G is supersolvable by Lemma 2.6 and consequently G = UW ,
where U is a normal nilpotent Hall subgroup of G of odd order, W is
a supersolvable Hall subgroup of G with (|U |, |W |) = 1 by Lemma 3.5.
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166 Characterization of finite groups
Since O2(N)R = O2(N) × R for any Sylow subgroup R of G of odd order,
it follows that O2(N) 6 Z∞(G) by [11, Theorem 6.3, p. 221]. Thus (2)
holds.
(2) =⇒ (3) Since G/U ≃ W is supersolvable and U is nilpotent, it
follows that G is solvable and so N is solvable. Let R be any Sylow
subgroup of G. If R 6 U , then R is normal in G. If (|R|, |U |) = 1,
then by (2), every non-cyclic Sylow subgroup P of F (N) of odd order
has a subgroup D such that 1 < |D| < |P | and all subgroups H of
P with |H| = |D| permute with R. Thus either R is normal in G or
(|R|, |U |) = 1, we have that every non-cyclic Sylow subgroup P of F (N)
of odd order has a subgroup D such that 1 < |D| < |P | and all subgroups
H of P with |H| = |D| are S-quasinormal in G. On the other hand,
O2(N) 6 Z∞(G). Then by [11, Theorem 6.2, p. 221], every subgroup of
O2(N) is S-quasinormal in G. Thus (3) holds.
(3) =⇒ (1) It is clear.
Corollary 3.6. Let G be a group of composite order such that G is
quaternion-free. Then the following statements are equivalent:
(1) Every non-cyclic Sylow subgroup P of F ∗(G) has a subgroup D
such that 1 < |D| < |P | and all subgroups H of P with |H| = |P |
are S-quasinormal in G.
(2) G = UW , where U is a normal nilpotent Hall subgroup of G of odd
order, W is a supersolvable Hall subgroup of G with (|U |, |W |) = 1,
every non-cyclic Sylow subgroup P of F (G) of odd order has a
subgroup D such that 1 < |D| < |P | and all subgroups H of P with
|H| = |D| permute with R, where R is any Sylow subgroup of G
with (|R|, |U |) = 1 and O2(G) 6 Z∞(G).
(3) G is solvable and every non-cyclic Sylow subgroup P of F (G) has
a subgroup D such that 1 < |D| < |P | and all subgroups H of P
with |H| = |D| are S-quasinormal in G.
Proof. This is an immediate consequence of Theorem 1.2 if N = G.
As an immediate corollary of Theorem 1.2 we get Theorem 1.1.
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M. Asaad, P. Csörgő 167
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Contact information
M. Asaad Cairo University, Faculty of Science, Department
of Mathematics,
Giza 12613, Egypt
E-Mail: moasmo45@hotmail.com
Piroska Csörgő Eötvös University, Department of Algebra and
Number Theory,
Pázmány Péter sétány 1/c,
H–1117 Budapest, Hungary
E-Mail: ska@cs.elte.hu
Received by the editors: 01.02.2012
and in final form 26.05.2012.
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