Finite groups with X-quasipermutable Sylow subgroups
Let H ≤ E and X be subgroups of a finite group G. Then we say that H is X-quasipermutable (XS-quasipermutable, respectively) in E provided that G has a subgroup B such that E = NE(H)B and H X-permutes with B and with all subgroups (with all Sylow subgroups, respectively) V of B such that (|H|, |V |)...
Saved in:
| Date: | 2015 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2015
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2104 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508033693515776 |
|---|---|
| author | Xiaolan, Yi Xue, Yang Сяолань, Ій Хуе, Янг |
| author_facet | Xiaolan, Yi Xue, Yang Сяолань, Ій Хуе, Янг |
| author_sort | Xiaolan, Yi |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:50:29Z |
| description | Let H ≤ E and X be subgroups of a finite group G. Then we say that H is X-quasipermutable (XS-quasipermutable,
respectively) in E provided that G has a subgroup B such that E = NE(H)B and H X-permutes with B and with all subgroups (with all Sylow subgroups, respectively) V of B such that (|H|, |V |) = 1. We analyze the influence of X-quasipermutable and XS-quasipermutable subgroups on the structure of G. In particular, it is proved that if every Sylow subgroup P of G is F(G)-quasipermutable in its normal closure PG in G, then G is supersoluble. |
| first_indexed | 2026-03-24T02:18:47Z |
| format | Article |
| fulltext |
UDC 512.542
Xiaolan Yi*, Xue Yang (Zhejiang Sci.-Techn. Univ., Hangzhou, China)
FINITE GROUPS WITH X-QUASIPERMUTABLE SYLOW SUBGROUPS
СКIНЧЕННI ГРУПИ З X-КВАЗIПЕРЕСТАВНИМИ
СИЛОВСЬКИМИ ПIДГРУПАМИ
Let H ≤ E and X be subgroups of a finite group G. Then we say that H is X-quasipermutable (XS-quasipermutable,
respectively) in E provided that G has a subgroup B such that E = NE(H)B and H X-permutes with B and with
all subgroups (with all Sylow subgroups, respectively) V of B such that (|H|, |V |) = 1. We analyze the influence of
X-quasipermutable and XS-quasipermutable subgroups on the structure of G. In particular, it is proved that if every Sylow
subgroup P of G is F (G)-quasipermutable in its normal closure PG in G, then G is supersoluble.
Нехай H ≤ E i X — пiдгрупи скiнченної групи G. Тодi говорять, що H є X-квазiпереставною (XS-квазiперестав-
ною, вiдповiдно) в E, якщо G мiстить таку пiдгрупу B, що E = NE(H)B i H є X-переставною з B i з усiма
пiдгрупами (з усiма силовськими пiдгрупами, вiдповiдно) V з B такими, що (|H|, |V |) = 1. У данiй роботi
проаналiзовано вплив X-квазiпереставних i XS-квазiпереставних пiдгруп на будову G. Зокрема, доведено, що
якщо кожна силовська пiдгрупа P iз G є F (G)-квазiпереставною в її нормальному замиканнi PG в G, то G є
надрозв’язною.
1. Introduction. Throughout this paper, all groups are finite and G always denotes a finite group.
For any prime p we use Cp to denote a group of order p.
If AB = BA, then A is said to permute with B; if G = AB, then B is called a supplement of A
to G; if ABx = BxA, for at least one element x ∈ X ⊆ G, then A is said to X-permute with B [1].
A large number of researches are connected with the study of subgroups H of G such that H
permutes with some subgroups of H’s supplement B in G. If, for example, H X-permutes with
all subgroups of B, then H is called X-semipermutable in G [2]; if H permutes with all Sylow
subgroups of B, then H is called SS-quasinormal in G [3]. Subgroups with a condition of such kind
have been useful in the analysis of many aspects of the theory of finite groups.
In this paper, we introduce and analyze some applications of the following concept that cover the
conditions of X-semipermutability and SS-quasinormality.
Definition 1.1. LetH ≤ E andX be subgroups ofG. Then we say thatH isX-quasipermutable
(XS-quasipermutable, respectively) in E provided G has a subgroup B such that E = NE(H)B and
H X-permutes with B and with all subgroups (with all Sylow subgroups, respectively) V of B such
that (|H|, |V |) = 1.
Example 1.1. Let p, q and r be different primes such that q divides p−1. Let A = CpoCq be a
non-Abelian group of order pq and R a simple FrA-module which is faithful for A. Let G = RoA.
Then Cp, clearly, is RS-quasipermutable in G. On the other hand, |R| > r when p > r and so Cp is
not R-quasipermutable in G.
It is clear that every X-semipermutable subgroup and every SS-quasinormal subgroup of G are
XS-quasipermutable in G for any X ⊆ G. We shall show that the inverse statements are not true in
general.
Example 1.2. Let p, q and r be different primes such that qr divides p− 1.
* Supported by the NNSF grant of China (Grant No. 11471055).
c© XIAOLAN YI, XUE YANG, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12 1715
1716 XIAOLAN YI, XUE YANG
(i) Let G = (Cp o Cq) × P, where Cp o Cq is a non-Abelian group of order pq and Cp = 〈a〉
and P = 〈b〉 are groups of order p. Then Cq is clearly quasipermutable in G, and for every x ∈ G,
〈ab〉xCq 6= Cq〈ab〉x. Thus Cq is not G-semipermutable in G.
(ii) Let G = Cp o (Cq × Cr), where Cq × Cr ≤ Aut(Cp). Then Cq is 1-quasipermutable in
G. Assume that Cq is SS-quasinormal in G. For any supplement B of Cq to G we have Cp ≤ B,
so for every 1 6= x ∈ Cp we have CqCxr = CxrCq, which implies that G = CGr ≤ NG(Cq), so
Cq ≤ CG(Cp). This contradiction shows that Cq is not SS-quasinormal in G.
Our main goal here is to prove the following results.
Theorem A. Let X = F (G) be the Fitting subgroup of G and H a Hall X-quasipermutable
subgroup of G. If p > q for all primes p and q such that p divides |H| and q divides |G : H|, then H
is normal in G.
Corollary 1.1 (see [1], Theorem 5.4). Let X = F (G) be the Fitting subgroup of G and H a
Hall X-semipermutable subgroup of G. If p > q for all primes p and q such that p divides |H| and
q divides |G : H|, then H is normal in G.
Corollary 1.2 (see [4], Theorem 3). If a Sylow p-subgroup P of G, where p is the largest prime
dividing |G|, is 1-semipermutable in G, then P is normal in G.
Theorem B. Let X = F (G) be the Fitting subgroup of G. If every Sylow subgroup P of G is
X-quasipermutable in its normal closure PG in G, then G is supersoluble.
Corollary 1.3. If every Sylow subgroup P of G is 1-semipermutable in its normal closure PG in
G, then G is supersoluble.
Note that if a subgroup H of G is 1-semipermutable in G, then H is 1-semipermutable in every
subgroup of G containing H. Hence we get from Corollary 1.3 the following known result.
Corollary 1.4 (see [4], Theorem 5). If every Sylow subgroup of G is 1-semipermutable in G,
then G is supersoluble.
From Theorem B we also get the following result.
Corollary 1.5 (see [4], Theorem 1.11). If every Sylow subgroup of G is F (G)-quasipermutable
in G, then G is supersoluble.
We use Mφ(G) to denote a set of maximal subgroups of G such that Φ(G) coincides with the
intersection of all subgroups in Mφ(G).
Theorem C. Let P be a Sylow p-subgroup of G and X = Op′,p(G). Suppose that every number
V of some fixed Mφ(P ) is XS-quasipermutable in G.
(i) If |P | > p, then G is p-supersoluble.
(ii) If (p− 1, |G|) = 1, then G is p-nilpotent.
Corollary 1.6 (see [3], Theorem 1.1). Let P be a Sylow p-subgroup of G, where p is the smallest
prime dividing |G|. If every number V of some fixed Mφ(P ) is SS-quasinormal in G, then G is
p-nilpotent.
Corollary 1.7. Let P be a Sylow p-subgroup of G and X = F (G). If NG(P ) is p-nilpotent and
every number V of some fixed Mφ(P ) is XS-quasipermutable in G, then G is p-nilpotent.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
FINITE GROUPS WITH X-QUASIPERMUTABLE SYLOW SUBGROUPS 1717
Proof. If |P | = p, then G is p-nilpotent by Burnside’s theorem [6] (IV, 2.6). Otherwise, G is
p-supersoluble by Theorem C. The hypothesis holds for G/Op′(G) (see Lemma 2.2 below) and so
in the case when Op′(G) 6= 1, G/Op′(G) is p-nilpotent by induction, which implies the p-nilpotency
of G. Therefore we may assume that Op′(G) = 1. But then, by Lemma 2.4(3) below, P is normal in
G. Hence G is p-nilpotent by hypothesis.
From Corollary 1.7 we get the following corollary.
Corollary 1.8 (see [3], Theorem 1.2). Let P be a Sylow p-subgroup ofG. IfNG(P ) is p-nilpotent
and every number V of some fixed Mφ(P ) is SS-quasinormal in G, then G is p-nilpotent.
2. Preliminaries. The first lemma is evident.
Lemma 2.1. Let A, B and X be subgroups of G and N a normal subgroup of G. If A X-
permutes with B, then AN/N (XN/N)-permutes with BN/N. Hence in the case when X ≤ N,
AN/N permutes with BN/N.
Lemma 2.2. Let H and X be subgroups of G and N a normal subgroup of G. Suppose that H
is X-quasipermutable (XS-quasipermutable, respectively) in G.
(1) If either H is a Hall subgroup of G or for every prime p dividing |H| and for every Sylow
p-subgroup Hp of H we have Hp � N, then HN/N is (XN/N)-quasipermutable
(
(XN/N)S-
quasipermutable, respectively
)
in G/N.
(2) If H is 1S-quasipermutable in G, then H permutes with some Sylow p-subgroup of G for all
primes p such that (|H|, p) = 1.
Proof. (1) By hypothesis there is a subgroup B of G such that G = NG(H)B and H X-
permutes with B and with all subgroups (with all Sylow subgroups, respectively) L of B such that
(|H|, |L|) = 1.
It is clear that G/N = NG/N (HN/N)(BN/N). Let K/N be any subgroup (any Sylow p-
subgroup, respectively) of BN/N such that (|HN/N |, |K/N |) = 1. ThenK = (K∩B)N. Let B0 be
a minimal supplement ofK∩B∩N inK∩B. ThenK/N = (K∩B)N/N = B0(K∩B∩N)N/N =
= B0N/N and K ∩ B ∩N ∩ B0 ≤ Φ(B0). Therefore π(K/N) = π(B0), so (|HN/N |, |B0|) = 1.
It follows that (|H|, |B0|) = 1, so in the case when H is X-quasipermutable in G, H X-permutes
with B0 and hence HN/N (XN/N)-permutes with K/N = B0N/N. Thus HN/N is (XN/N)-
quasipermutable in G/N.
Finally, suppose that H is XS-quasipermutable in G and K/N is a Sylow p-subgroup of BN/N.
Then B0 is a p-group, so (|H|, p) = 1 and for some Sylow p-subgroup Bp of B we have B0 ≤ Bp.
ThenK/N = B0N/N and henceHN/N (XN/N)-permutes withK/N. ThusHN/N is (XN/N)S-
quasipermutable in G/N.
(2) By [6] (VI, 4.6), there are Sylow p-subgroups P1, P2 and P of NG(H), B and G, respectively,
such that P = P1P2. Hence H permutes with P.
Lemma 2.3. Let A and B be subgroups of G such that G = AB. Then G = ABx for all x ∈ G.
Proof. Let x = ab, where a ∈ A and b ∈ B. Then ABx = ABab = AabBb−1a−1 = ABa−1 =
= Ga−1 = G.
We shall need in our proofs the following properties of p-supersoluble groups.
Lemma 2.4. (1) If G/Φ(G) is p-supersoluble, then G is p-supersoluble [6] (IV, 8.6).
(2) LetN andR be distinct minimal normal subgroups ofG. IfG/N andG/R are p-supersoluble,
then G is p-supersoluble.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
1718 XIAOLAN YI, XUE YANG
(3) Let A = G/Op′(G). Then G is p-supersoluble if and only if A/Op(A) is an Abelian group of
exponent dividing p − 1, p is the largest prime dividing |A| and F (A) = Op(A) is a normal Sylow
subgroup of A.
Proof. (2) This follows from the G-isomorphism NR/N ' R.
(3) Since G is p-supersoluble if and only if G/Op′(G) is p-supersoluble, we may assume without
loss of generality that Op′(G) = 1.
First assume that G is p-supersoluble. In this case G/CG(H/K) is an Abelian group of exponent
dividing p− 1 for any chief factor H/K of G of order divisible by p. On the other hand,
Op′,p(G) = Op(G) = ∩
{
CG(H/K) | H/K is a chief factor of G and p ∈ π(H/K)
}
by [8] (A, 13.2). Hence G/Op(G) is an Abelian group of exponent dividing p − 1. Thus p is the
largest prime dividing |G| and F (G) = Op(G) is a normal Sylow p-subgroup of G.
Finally, if G/Op(G) is an Abelian group of exponent dividing p−1, then every chief factor H/K
of G below Op(G) is cyclic by [8] (B, 9.8(d)). Hence G is supersoluble.
Lemma 2.5 [7]. If G has three nilpotent subgroups A1, A2 and A3 whose indices |G : A1|,
|G : A2|, |G : A3| are pairwise coprime, then G is itself nilpotent.
Lemma 2.6. Let G = P o E, where P is the Sylow p-subgroup of G and E is a Sylow tower
group. Suppose that for every Sylow subgroup Q of E there is a subgroup B of P such that
P = NP (Q)B and Q permutes with all subgroups of B. Then G is p-supersoluble.
Proof. Suppose that this lemma is false and let G be a counterexample of minimal order. It is
clear that G is soluble and |P | > p. Let p1 > . . . > pt be the set of all prime divisors of |E|. Let Pi
be a Sylow pi-subgroup of E.
Let N be a normal subgroup of G. Then the hypothesis holds for G/N, so the choice of G and
Lemma 2.4 imply that N is the only minimal normal subgroup of G and N � Φ(G). Therefore
N = CG(N) = F (G) = P by [8] (A, 15.2), so E is a maximal subgroup of G.
Assume that |π(E)| > 2. Then t > 2. Let Ei be a Hall p′i-subgroup of E. Then the hypothesis
holds for PEi, so PEi is p-supersoluble by the choice of G. Moreover, since P = CG(P ) we
have Op′(PEi) = 1. Therefore PEi is supersoluble by Lemma 2.4(3), and F (PEi) = P. Thus
PEi/P ' Ei is an Abelian group of exponent dividing p − 1. Therefore E has at least three
Abelian subgroups Ei, Ej and Ek of exponent dividing p − 1 whose indices |E : Ei|, |E : Ej |,
|E : Ek| are pairwise coprime. But then by Lemma 2.5, E is nilpotent, and every Sylow subgroup of
E is an Abelian group of exponent dividing p−1. Hence E is an Abelian group of exponent dividing
p− 1, which implies that |P | = p. This contradiction shows that |π(E)| = 2.
SinceE is a Sylow tower group, P1 is normal inE and soNG(P1)∩P = 1. Therefore P1 permutes
with all subgroups of P. If P ≤ NG(P2), then PP2 = P ×P2. Hence in this case P2 ≤ CG(P ) = P.
This contradiction shows that NG(P2) ∩ P 6= P, so there is a nonidentity subgroup B < P such that
P2B = BP2. Hence BE = B(P1P2) = (P1P2)B = BE is a subgroup of G, which contradicts the
maximality of E = P1P2.
Lemma 2.7 (see [9], Theorem E). Suppose that G = AB and P ≤ Op(A). Assume that every
conjugate of P in A permutes with every Sylow q-subgroup of B for all primes q 6= p. Then PG is
soluble and the p-complements in PG are nilpotent.
Lemma 2.8 (see [10], Lemma 2.15). Let E be a normal nonidentity quasinilpotent subgroup of
G. If Φ(G) ∩ E = 1, then E is the direct product of some minimal normal subgroups of G.
Lemma 2.9. Let H be a subnormal subgroup of G. If H is nilpotent, soluble, or a π-group,
then HG is nilpotent, soluble, or a π-group, respectively.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
FINITE GROUPS WITH X-QUASIPERMUTABLE SYLOW SUBGROUPS 1719
Proof. See the proof of Theorem 2.2 in [11, Ch. 2].
3. Proofs of the results. Proof of Theorem A. Suppose that this theorem is false and let G
be a counterexample of minimal order. Let π be the set of all prime divisors of H. By hypothesis,
there is a subgroup B of G such that G = NG(H)B and H X-permutes with B and with every
π′-subgroup of B. Let x ∈ X such that HBx = BxH. Then 〈H,Bx〉 = HBx and G = NG(H)Bx
by Lemma 2.3. Therefore HG = HNG(H)Bx
= HBx ≤ HBx. Hence HG = H(HG ∩Bx).
(1) HN is normal in G for any nonidentity normal subgroup N of G. Hence Oπ(G) = 1.
It is clear that HN/N is a Hall π-subgroup of G/N and the hypothesis holds for (G/N,HN/N)
by Lemma 2.2. Hence HN/N is normal in G by the choice of G. Thus HN is normal in G. Since
Oπ(G) ≤ H, it follows that in the case when Oπ(G) 6= 1, Oπ(G)H = H is normal in G, contrary
to the choice of G. Hence we have (1).
(2) F (G) is a π′-group.
Since Oπ(F (G)) is characteristic in F (G), it is normal in G. Hence by (1), Oπ(F (G)) ≤
≤ Oπ(G) = 1.
(3) F (G) = Op(G) for some prime p 6∈ π.
Let p be a prime dividing |F (G)| and P the Sylow p-subgroup of F (G). Then by claim (2),
p 6∈ π. Suppose that P 6= F (G). Then F (G) = P × E, where E 6= 1 is the Hall p′-subgroup of
F (G). Since P and E are characteristic in F (G), both these subgroup are normal in G. But then
HP and HE are normal in G by claim (1), so H = HP ∩HE is normal in G. This contradiction
shows that F (G) = P.
(4) F (G) is an elementary Abelian p-group.
Assume that this is false. Then Φ(F (G)) 6= 1. Since Φ(F (G)) is characteristic in F (G), it is
normal in G. Hence by claim (1), Φ(F (G))H is normal in G. But Φ(F (G))H is π-soluble and so
any two Hall π-subgroups of Φ(F (G))H are conjugate in Φ(F (G))H. Therefore, by the Frattini
argument, G = (Φ(F (G))H)NG(H) = Φ(F (G))NG(H) = NG(H) since Φ(F (G)) ≤ Φ(G), a
contradiction. Hence we have (4).
(5) G 6= HB.
Suppose thatG = HB.Without loss of generality we may assume thatB is a minimal supplement
of H in G. First assume that H permutes with all π′-subgroups of B. Then the hypothesis holds
for every subgroup of G containing H. Therefore for every maximal subgroup V of B we have
V ≤ NG(H) by the choice of G, so V is the only maximal subgroup of B. Hence B is a cyclic
group of order qn for some prime q. It is clear that q is the smallest prime dividing |G| and, in view
of claim (1), (H ∩B)G = (H ∩B)HB = (H ∩B)H ≤ HG = 1. Hence H ∩B = 1. Therefore |G :
HV | = q, which implies that HV is normal in G. But then, since V ≤ NG(H), H is normal in
G. This contradiction shows that for some π′-subgroup A of B we have HA 6= AH. It follows that
F (G) 6= 1. Moreover, since G = HB, F (G) ≤ B by claim (3). Hence by claim (4), the hypothesis
holds for (HF (G), H). Therefore, if HF (G) 6= G, then H is normal (and so characteristic) in
HF (G). Hence in this case H is normal in G by claim (1). Thus HF (G) = G and so the minimality
of B implies that B = F (G). But then, by claim (4), HA = AH. This contradiction shows that we
have (5).
(6) H permutes with every subgroup of B ∩Op(G) (this directly follows from claim (4)).
(7) Op(G) = 1.
Suppose that F (G) = Op(G) 6= 1. Then:
(a) Op(G)NG(H) = G.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
1720 XIAOLAN YI, XUE YANG
By claim (1), HOp(G) is normal in G. On the other hand, HOp(G) is p-soluble and so any
two Hall π-subgroups of HOp(G) are conjugate in HOp(G). Therefore, by the Frattini argument,
G = (HOp(G))NG(H) = Op(G)NG(H).
(b) HG = H(HG ∩Op(G)).
In view of (a) we have
HG = HOp(G)NG(H) = HOp(G) ≤ HOp(G),
so HG = HG ∩HOp(G) = H(HG ∩Op(G)).
(c) HG ∩Op(G) is a subgroup of B.
HG = H(HG ∩Bx) = H(HG ∩Op(G)) by (b). Hence HG ∩Op(G) ≤ B by claim (3).
Final contradiction for (7). In view of claims (6), (b) and (c), the hypothesis holds for HG.
Hence in the case when HG 6= G, H is normal in HG, which implies the normality H in G. Thus
HG = G. But then G = HG = H(HG ∩Bx) = HBx = HB, which contradicts (5).
Final contradiction. Since X = F (G) = Op(G) = 1 by claim (7), the hypothesis holds for
HG = H(HG ∩B) ≤ G. Hence HG = G, which implies that G = HB, contrary to (5).
The theorem is proved.
Proof of Theorem B. Suppose that this theorem is false and let G be a counterexample with |G|
minimal. Let R be a minimal normal subgroup of G. Then X/X ∩R ' XR/R ≤ F (G/R).
(1) The hypothesis holds for G/R. Hence G/R is supersoluble.
Let P be a Sylow p-subgroup of G and D = PG. Suppose that P � R. By hypothesis,
D = ND(P )B, where B is a subgroup of D such that P X-permutes with B and with all p′-
subgroups of B. Then
(PR/R)G/R = (PR)G/R = PGR/R =
= DR/R = (ND(P )R/R)(BR/R) = NDR/R(PR/R)(BR/R)
and PR/R (XR/R)-permutes with BR/R by Lemma 2.1.
Now, let V/R ≤ BR/R, where (p, |V/R|) = 1. Let U be a minimal supplement to R in V. Then
U ∩R ≤ Φ(U), so (p, |U |) = 1. Then for some x ∈ X we have PUx = UxP, so
(PR/R)(UR/R)xR = (PR/R)(V/R)xR = (V/R)xR(PR/R),
where xR ∈ XR/R ≤ F (G/R). Therefore PR/R is F (G/R)-quasipermutable in (PR/R)G/R, so
the hypothesis holds for G/R. Thus G/R is supersoluble by the choice of G.
(2) G is soluble.
If X 6= 1, this follows from claim (1). Now assume that X = 1. Let p be the largest prime
dividing |G| and P a Sylow p-subgroup of G. Then P is normal, and so, characteristic in PG by
Theorem A. Hence P is normal in G and so P ≤ X, a contradiction.
(3) R = X = CG(R) = Op(G) for some prime p, and G = RoM, where M is a supersoluble
maximal subgroup of G.
Claim (1) and Lemma 2.4 imply that R is the unique minimal normal subgroup of G and
R � Φ(G), so CG(R) ≤ R. Thus we have (3) by claims (1), (2) and [8] (A, 17.2).
(4) p is the largest prime dividing.
Assume that this is false. Let q be the largest prime dividing |G| and Q a Sylow q-subgroup of
M. Then D = QG = R o Q by claims (1) and (3). Moreover, NG(Q) = M by claim (3). Hence
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
FINITE GROUPS WITH X-QUASIPERMUTABLE SYLOW SUBGROUPS 1721
ND(Q) = Q. By hypothesis and claim (3), there is a subgroup B of D such that D = QB and Q
R-permutes with all p-subgroups of B. But, clearly, R ≤ B. Hence Q is X-quasipermutable in D,
so Q is normal in D by Theorem A. That implies that Q ≤ CG(R), contrary to (3).
(5) R is a Sylow p-subgroup of G (this directly follows from claims (1), (3) and (4)).
Final contradiction. Let Q be any Sylow subgroup of M. Then Q is a Sylow subgroup of G and
so, by hypothesis and claim (3), there is a subgroup B of G such that QG = NQG(Q)B and Q R-
permutes with every p-subgroup of B. It is clear that R = (R∩NQG(Q))(R∩B) = NR(Q)(R∩B).
Therefore G is p-supersoluble by Lemma 2.6, which implies that |N | = p. This contradiction
completes the proof of the result.
Proof of Theorem C. (i) Suppose that this assertion is false and let G be a counterexample of
minimal order.
Let V ∈Mφ(P ) andD = V G. By hypothesis, there is a subgroupB ofG such thatG = NG(V )B
and V is X-permutable with B and with all Sylow subgroups S of B such that (p, |S|) = 1.
(1) Op′(N) = 1 for every subnormal subgroup N of G. Hence X ≤ Op(G).
Indeed, suppose that for some subnormal subgroup N of G we have Op′(N) 6= 1. Then Op′(G) 6=
6= 1 by Lemma 2.9, and the hypothesis holds for G/Op′(G) by Lemma 2.2. Hence G/Op′(N) is
p-supersoluble by the choice of G. Thus G is p-supersoluble, a contradiction. Therefore Op′(N) = 1.
Therefore, since X is p-nilpotent, X ≤ Op(G).
(2) If L is a minimal normal subgroup of G, then L � Φ(P ).
Indeed, in the case when L ≤ Φ(P ), we have L ≤ Φ(G) and the hypothesis holds for G/L by
Lemma 2.2. Hence G/L is p-supersoluble by the choice of L. Therefore G is p-supersoluble by
Lemma 2.4(1), a contradiction.
(3) D is soluble, so Op(G) 6= 1.
Assume that Op(G) = 1. Then in view of claim (1), X = 1. Therefore V permutes with B
and with all Sylow subgroups S of B such that (p, |S|) = 1. Therefore D = V G = V NG(V )B =
= V B ≤ V B, so D = V (D ∩ B). Hence V D is soluble by Lemma 2.7. But claim (1) implies that
Op′(V
D) = 1. Hence Op(V D) 6= 1, and Op(V D) ≤ Op(G) by Lemma 2.9. Thus Op(G) 6= 1, a
contradiction.
(4) P is not cyclic.
Assume that P is cyclic. Claim (3) implies that for some minimal normal subgroup L of G we
have L ≤ Op(G) ≤ P. Then |L| = p, and since L � Φ(P ) by claim (2), we get L = P, contrary to
the hypothesis.
(5) Every normal p-soluble subgroup of G is supersoluble and p-closed (see claim (5)(a) in the
proof of Proposition in [12]).
(6) G is not p-soluble (this directly follows from claim (5)).
Final contradiction for (i). In view of claim (4), there is a subgroup W ∈ Mφ(P ) such that
V 6= W. Then P = VW. In view of claims (3) and (6), P � D. Hence V is a Sylow subgroup of D,
so V is normal in D (and also in G) by claim (5). Similarly, W is normal in G. Hence P is normal
in G, contrary to claim (6). This final contradiction completes the proof of assertion (i).
(ii) If |P | = p, then G is p-nilpotent by [6] (IV, 2.6). Let |P | > p and H/K any chief factor of G
of order divisible by p. Then |H/K| = p by assertion (i), so CG(H/K) = G since (p− 1, |G|) = 1.
Hence G is p-nilpotent.
The theorem is proved.
1. Guo W., Skiba A. N., Shum K. P. X-permutable subgroups of finite groups // Sib. Math. J. – 2007. – 48, № 4. –
P. 593 – 605.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
1722 XIAOLAN YI, XUE YANG
2. Guo W., Shum K. P., Skiba A. N. X-semipermutable subgroups of finite groups // J. Algebra. – 2007. – 215. –
P. 31 – 41.
3. Li S., Shen Z., Liu J., Liu X. The influence of SS-quasinormality of some subgroups on the structure of finite group //
J. Algebra. – 2008. – 319. – P. 4275 – 4287.
4. Podgornaya V. V. Seminormal subgroups and supersolubility of finite groups // Vesti NAN Belarus. Ser. Phys.-Math.
Sci. – 2000. – 4. – P. 22 – 26.
5. Yi X., Yang X. Finite groups with X-quasipermutable subgroups of prime power order // Bull. Iran. Math. Soc. (to
appear).
6. Huppert B. Endliche Gruppen I. – Berlin etc.: Springer-Verlag, 1967. – 793 p.
7. Kegel O. H. Zur Struktur mehrfach faktorisierbarer endlicher Gruppen // Math. Z. – 1965. – 87. – S. 409 – 434.
8. Doerk K., Hawkes T. Finite soluble groups. – Berlin; New York: Walter de Gruyter, 1992. – 893 p.
9. Isaacs I. M. Semipermutable π-subgroups // Arch. Math. – 2014. – 102. – P. 1 – 6.
10. Guo W., Skiba A. N. On FΦ∗-hypercentral subgroups of finite groups // J. Algebra. – 2012. – 372. – P. 285 – 292.
11. Isaacs I. M. Finite group theory // Grad. Stud. Math. – Providence, RI: Amer. Math. Soc., 2008. – 92.
12. Yi X., Skiba A. N. Some new characterizations of PST -groups // J. Algebra. – 2014. – 399. – P. 39 – 54.
Received 09.06.14,
after revision — 11.03.15
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 12
|
| id | umjimathkievua-article-2104 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:18:47Z |
| publishDate | 2015 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/94/9efb5599ab2f6beed3d3cc61afa82294.pdf |
| spelling | umjimathkievua-article-21042019-12-05T09:50:29Z Finite groups with X-quasipermutable Sylow subgroups Скiнченнi групи з X-квазiпереставними силовськими пiдгрупами Xiaolan, Yi Xue, Yang Сяолань, Ій Хуе, Янг Let H ≤ E and X be subgroups of a finite group G. Then we say that H is X-quasipermutable (XS-quasipermutable, respectively) in E provided that G has a subgroup B such that E = NE(H)B and H X-permutes with B and with all subgroups (with all Sylow subgroups, respectively) V of B such that (|H|, |V |) = 1. We analyze the influence of X-quasipermutable and XS-quasipermutable subgroups on the structure of G. In particular, it is proved that if every Sylow subgroup P of G is F(G)-quasipermutable in its normal closure PG in G, then G is supersoluble. Нехай H ≤ E i X — пiдгрупи скiнченної групи G. Тодi говорять, що H є X-квазiпереставною (XS-квазiпереставною, вiдповiдно) в E, якщо G мiстить таку пiдгрупу B, що E = NE(H)B i H є X-переставною з B i з усiма пiдгрупами (з усiма силовськими пiдгрупами, вiдповiдно) V з B такими, що (|H|, |V |) = 1. У данiй роботi проаналiзовано вплив X-квазiпереставних i XS-квазiпереставних пiдгруп на будову G. Зокрема доведено, що якщо кожна Силовська пiдгрупа P iз G F(G)-квазiпереставна в його нормальному замиканнi PG в G, то G є надрозв’язною. Institute of Mathematics, NAS of Ukraine 2015-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2104 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 12 (2015); 1715-1722 Український математичний журнал; Том 67 № 12 (2015); 1715-1722 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2104/1212 Copyright (c) 2015 Xiaolan Yi; Xue Yang |
| spellingShingle | Xiaolan, Yi Xue, Yang Сяолань, Ій Хуе, Янг Finite groups with X-quasipermutable Sylow subgroups |
| title | Finite groups with X-quasipermutable Sylow subgroups |
| title_alt | Скiнченнi групи з X-квазiпереставними силовськими пiдгрупами |
| title_full | Finite groups with X-quasipermutable Sylow subgroups |
| title_fullStr | Finite groups with X-quasipermutable Sylow subgroups |
| title_full_unstemmed | Finite groups with X-quasipermutable Sylow subgroups |
| title_short | Finite groups with X-quasipermutable Sylow subgroups |
| title_sort | finite groups with x-quasipermutable sylow subgroups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2104 |
| work_keys_str_mv | AT xiaolanyi finitegroupswithxquasipermutablesylowsubgroups AT xueyang finitegroupswithxquasipermutablesylowsubgroups AT sâolanʹíj finitegroupswithxquasipermutablesylowsubgroups AT hueâng finitegroupswithxquasipermutablesylowsubgroups AT xiaolanyi skinčennigrupizxkvaziperestavnimisilovsʹkimipidgrupami AT xueyang skinčennigrupizxkvaziperestavnimisilovsʹkimipidgrupami AT sâolanʹíj skinčennigrupizxkvaziperestavnimisilovsʹkimipidgrupami AT hueâng skinčennigrupizxkvaziperestavnimisilovsʹkimipidgrupami |