$s$-Conditionally Permutable Subgroups and $p$-Nilpotency of Finite Groups
We study the $p$-nilpotency of a group such that every maximal subgroup of its Sylow $p$-subgroups is $s$-conditionally permutable for some prime $p$. By using the classification of finite simple groups, we get interesting new results and generalize some earlier results.
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| author | Li, X. H. Xu, Y. Лі, Х. Х. Ху, Я. |
| author_facet | Li, X. H. Xu, Y. Лі, Х. Х. Ху, Я. |
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| description | We study the $p$-nilpotency of a group such that every maximal subgroup of its Sylow $p$-subgroups is $s$-conditionally permutable for some prime $p$. By using the classification of finite simple groups, we get interesting new results and generalize some earlier results. |
| first_indexed | 2026-03-24T02:20:18Z |
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UDC 512.5
Y. Xu (School Math. and Statist., Henan Univ. Sci. and Technol., China),
X. H. Li (School Math. Sci., Soochow Univ., China)
s-CONDITIONALLY PERMUTABLE SUBGROUPS
AND p-NILPOTENCY OF FINITE GROUPS*
s-УМОВНО ПЕРЕСТАВНI ПIДГРУПИ
ТА p-НIЛЬПОТЕНТНIСТЬ СКIНЧЕННИХ ГРУП
We study the p-nilpotency of a group such that every maximal subgroup of its Sylow p-subgroups is s-conditionally
permutable for some prime p. By using the classification of finite simple groups, we get interesting new results and
generalize some earlier results.
Вивчено p-нiльпотентнiсть групи, для якої кожна максимальна пiдгрупа її силовських p-пiдгруп є s-умовно пе-
реставною для деякого простого p. За допомогою класифiкацiї скiнченних простих груп отримано цiкавi новi
результати та узагальнено деякi результати, що отриманi ранiше.
1. Notation and introduction. In this paper, all groups are finite and G stands for a finite group.
Let π(G) be the set of all prime divisors of |G|. Let Gp and Sylp(G) be a Sylow p-subgroup and the
set of Sylow p-subgroups of G respectively. Let F denote a formation, U the class of supersolvable
groups. Let np be the p-part of a nature number n, that is, np = pa such that pa | n but pa+1 - n.
Let G be a Lie-type simple group over the finite field Fq. To collect some useful information and
for convenience in narrating, we define n(G) in Table 1.1. The other notation and terminology are
standard (see [11, 13]).
Table 1.1
G n(G) G n(G)
An(q) (n+ 1)f 2A2k(q)(k ≥ 2) (4k + 2)f
Bn(q)(p 6= 2) 2nf Bn(2f ) 2nf
Cn(q)(p 6= 2) 2nf 2A2k+1(q)(k ≥ 2) 2(k + 1)f
2Dn(q) 2nf Dn(q) 2(n− 1)f
E8(q) 30f E7(q) 18f
E6(q) 12f 2E6(q) 18f
F4(q) 12f 2F4(q)
′
12f
G2(q) 6f 3D4(q) 12f
2G2(q) 6f 2B2(q) 4f
Many authors have investigated the structure of a group when maximal subgroups of Sylow
subgroups of the group are well situated in the group. Srinivasan [28] showed that a group G is
supersolvable if all maximal subgroups of every Sylow subgroup of G are normal. Later, several
authors obtain the same conclusion if normality is replaced by some weaker property (see [25, 27]).
* This work was supported by the National Natural Science Foundation of China (Grant N.11171243, 11326056), the
Scientific Research Foundation for Doctors, Henan University of Science and Technology (N.09001610).
c© Y. XU, X. H. LI, 2014
858 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6
s-CONDITIONALLY PERMUTABLE SUBGROUPS AND p-NILPOTENCY OF FINITE GROUPS 859
In particular, these results indicate that the generalized normality of some maximal subgroups of
Sylow subgroups give a lot of useful information on the structure of groups.
In this paper, we obtain some sufficient conditions on p-nilpotency and supersolvability of groups
by using the s-conditional permutability of maximal subgroups of Sylow subgroups. Some earlier
results on this topic are generalized.
2. Basic definitions and preliminary results. Let H and K be two subgroups of G. We say
that H permutes with K if HK = KH . Recently, Huang and Guo [10] introduced a new embedding
property, namely, the s-conditional permutability of subgroups of a group.
Definition. A subgroup H of G is s-conditionally permutable if for every prime p ∈ π(G), there
exists a Sylow p-subgroup P of G such that HP = PH .
For the sake of convenience, we list here some known results which will be useful in the sequel.
Lemma 2.1 ([10], Lemma 2.3). Let H and K be subgroups of G. Then the following hold:
(1) IfH is s-conditionally permutable inG andK is normal inG, thenHK/K is s-conditionally
permutable in G.
(2) If H ≤ K � G and H is s-conditionally permutable in G, then H is s-conditionally
permutable in K.
Lemma 2.2 ([24], Lemma 6). Suppose that G is a non-Abelian simple group. Then there exists
an odd prime r ∈ π(G) such that G has no Hall {2, r}-subgroup.
Lemma 2.3 ([29], Theorem 3.1). Let F be a saturated formation containing U , and G a group
with a normal subgroup N such that G/N ∈ F . If all Sylow subgroups of F ∗(N) are cyclic, then
G ∈ F .
Lemma 2.4 ([26], Lemma 1.6). Let P be a nilpotent normal subgroup of a group G. If P ∩
∩ Φ(G) = 1, then P is the direct product of some minimal normal subgroups of G.
Recall that a prime divisor d of am − 1 is called primitive, if d does not divide ai − 1
for 1 ≤ i ≤ m − 1. For primitive prime divisors, an important property is due to Zsigmondy,
refer to [8].
Lemma 2.5 [8]. Let b and n be positive integers.
(1) There are primitive prime divisors of bn − 1 unless (b, n) = (2, 6) or b is a Mersenne prime
and n = 2.
(2) Each primitive prime divisor p of bn − 1 is at least n + 1. Moreover, if p = n + 1, then p2
divides bn − 1 except for the following cases:
(i) n = 2 and b = 2s − 1 or 3 · 2s − 1;
(ii) b = 2 and n = 4, 6, 10, 12 or 18;
(iii) b = 3 and n = 4 or 6;
(iv) b = 5 and n = 6.
(3) For a positive integer s, if a primitive prime divisor of bs−1 divides bn−1, then s divides n.
3. Main results and their proofs.
Theorem 3.1. Let G be a non-Abelian simple group and |G|2 = 2t. If G has a subgroup of
order 2t−1|G|r for every r ∈ π(G) \ {2}, then G ∼= PSL2(q), where q is a power of an odd prime
and t = 2.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6
860 Y. XU, X. H. LI
Proof. Let r ∈ π(G) \ {2}, H be a subgroup of G of order 2t−1|G|r, A ∈ Syl2(H) and
R ∈ Sylr(H). Then |A| = 2t−1 and R ∈ Sylr(G) and H = AR. Let M be a maximal subgroup
of G containing H . Then |M |2 = 2t or |M |2 = 2t−1. If |M |2 = 2t−1, then A ∈ Syl2(M) and H
is a Hall {2, r}-subgroup of M ; if |M |2 = 2t, then M2 ∈ Syl2(G), |G : M | is odd and so G has
a faithful primitive permutation representation of odd degree and M is listed in [20] (Theorem). By
the classification of finite simple groups, we divide the argument into the following cases.
(1) G is a sporadic simple group.
Let r = maxπ(G). Then by [5] and http://brauer.maths.qmul.ac.uk/Atlas/v3, 2t−1 - |M |, a
contradiction.
(2) G is an alternating An.
We have 2t =
(
1
2
n!
)
2
. Let r = maxπ(G). By [3], RA = R and 2t−1
∣∣∣∣12(r − 1)(n− r)! ,
this is impossible.
(3) G is a Lie-type simple group over GF (q), where q = pf and p is a prime.
Suppose that G = PSL2(q) and |G|2 > 4. If q = 2f , then G has no subgroup of order
1
2
|G|2|R| by [14], a contradiction. Hence q = pf with p odd. Thus (q − 1)2 = 2 or (q + 1)2 = 2.
If (q + 1)2 = 2, let t = maxπ(q + 1) and V ∈ Sylt(G), then G has no subgroup of order
1
2
|G|2|V |
by [14]; if (q − 1)2 = 2, let u = maxπ(q − 1) and U ∈ Sylu(G), then G has no subgroup of
order
1
2
|G|2|U | by [14], a contradiction. Hence |G|2 = 22, the result holds. From now, we assume
that n(G) > 2f .
Assume that (n(G), p) = (6, 2). Then (n(G)/f, f) is one of (3, 2) and (6, 1), and so G is one
of the groups PSL3(2
2), PSU4(2), PSL6(2), D4(2). Suppose that G ∈ {PSL3(2
2), PSU4(2),
D4(2)}. Let r = 3. Since Mr ∈ Sylr(G), by [5, p. 23, 26, and 85], M ∈ {A6, 32 · Q8} if
G = PSL3(2
2), M ∈ {31+2
+ : 2A4, 33 · S4} if G = PSU4(2) and M = 34 : 23 · S4 if
G = D4(2), hence 4 | |G : M |, a contradiction. Suppose that G = PSL6(2). Let r = 7.
By http://brauer.maths.qmul.ac.uk/Atlas/lin/L62, M ∈ {29 : (L3(2) × L3(2)), (L3(2) × L3(2)) : 2,
(L2(8) × 7) : 3}. If M ∈ {(L3(2) × L3(2)) : 2, (L2(8) × 7) : 3}, then 4 | |G : M |; if M =
= 29 : (L3(2) × L3(2)), since the maximal subgroup A of L3(2) satisfying 7 | |A| is isomorphic to
7 : 3, M has no the maximal subgroup of order 214 · 72, a contradiction. Hence (n(G), p) 6= (6, 2).
By Lemma 2.5, pn(G)−1 has at least one primitive prime divisor. Let r be the largest primitive prime
divisor of pn(G) − 1 and M a maximal subgroup of G of order 2t−1|G|r. Then M is not a parabolic
subgroup of G.
Suppose that G ∈ {PSL3(q), PSU3(q), 2F4(2
2m+1), Sz(q), 3D4(q), D4(2
f ), 2G2(q), G2(q)}.
The maximal subgroups or orders of maximal subgroups of 2B2(2
2m+1), PSL3(q) and PSU3(q) are
listed in the proof of Lemmas 1 – 4 in [7]; the maximal subgroups of 2F4(2
2m+1), 2G2(q), G2(2
f ),
3D4(q) and D4(2
f ) are listed in [6, 15 – 17, 23]. A simple checking shows that 4 | |G : M |, a
contradiction. Suppose that G = G2(q) with q odd. Since |Mr| = |G|r, by [16], the possibilities of
M are SL3(q) : 2, SU3(q) : 2, L2(13), G2(2) and J1. It is easy to prove that if M ∈ {SL3(q) : 2,
SU3(q) : 2, L2(13), G2(2), J1}, then M has no the subgroup of order 2t−1|G|r.
Next, we deal with the remaining Lie-type simple group G in the previous argument. Let H
be maximal subgroups of G containing a subgroup of G of order 2t−1|G|p. Then H is a parabolic
subgroup of G.
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s-CONDITIONALLY PERMUTABLE SUBGROUPS AND p-NILPOTENCY OF FINITE GROUPS 861
Suppose that G is an exceptional Lie-type simple group and the notation K(G) is defined in
[21] (Theorem). Suppose that p = 2. It is easy to see that the maximal subgroup in Table 1 [21]
don’t contain a subgroup of order 2t−1|G|r. Thus by [21] (Theorem), |M | < 2K(G)f . On the other
hand, by [12], |M | > (|M |2)2 ≥ 22(K(G)−1)f if G 6= E8(2
f ) or |M | > (|M |2)2 ≥ 22(K(G)−10)f
if G = E8(2
f ), a contradiction. Suppose that p > 2 and G is one of simple groups F4(q), E6(q),
2E6(q), E7(q), E8(q). Then 4 | |G : H|, this is impossible. Thus we have proved that there is no
exceptional Lie-type simple group satisfying the condition of Theorem 3.1.
Suppose that G is a classical simple group on n-dimension vector space V and n > 3. We
shall use the notation of the book [13] in the following argument. Aschbacher [1] classified maximal
subgroups of a classical simple group into 9 types: Ci, where 1 ≤ i ≤ 8, and S, see [13] for the
description.
Suppose that p = 2. If 3 < n < 12, using [14] and [15], it is easy to see that 4 | |G : M |, G
doesn’t satisfy the condition of Theorem 3.1. Hence we assume that n ≥ 12. Assume that M is an
almost simple group. Since 2t−1 | |M |2, by [18], |M | < 23fn < 2
1
2
n(n−2)f−2 ≤ (|M |2)2. On the
other hand, by [12], |M | > (|M |2)2, a contradiction. Suppose that M is a Ci subgroup. By [15]
(Table A – E), a simple checking shows that 4 | |G : M |, G doesn’t satisfy the hypothesis.
Assume that p > 2. Since 4 - |G : K|, we have 4 - n if G = PSLn(q); 2 - n if G = PSLn(q)
with 4 | (q + 1); 4 - (q + 1) if G 6= PSLn(q); 4 - n(n − 1) if G ∈ {Un(q), PSpn(q)}; 2 - k if
G ∈ {PΩ+
2k(q), PΩ2k+1(q)}; 2 - (k− 1) if G = PΩ−2k(q). Suppose that 2 < n < 12. From [14] and
[15], it is easy to see that either 2t−1 - |M | or Mr 6∈ Sylr(G), a contradiction. Hence we may assume
that n ≥ 12. By Lemma 2.5, we may assume that r > n(G) + 1 or r = n(G) + 1 and r2 | pn(G)− 1.
By [20], it is easy to see that |G : M | is not odd, hence M has a Hall {2, r}-subgroup. Suppose that
M is a S subgroup of G. Then the covering group of M is a subgroup of GLn(q) and there is a
non-Abelian simple group S such that S ≤ M ≤ Sut(S). Moreover, if N is the preimage of S in
G, then N is absolutely irreducible on V and N is not a classical group defined over a subfield of
GF (q)(in its natural representation). All possibilities of S have given in Examples 2.6 – 2.9 in [9].
For all possible S either 2t−1 - |M | or r2 - |M | when r = n(G) + 1, this is impossible. Suppose that
M is not a S subgroup of G. Since r | |M |, by [14] (Table 3.5.A – F), it is easy to see that M must
be one of C3, C6 and C8 subgroups of G. Since r > n(G) + 1 or r2 | |M | if r = n(G) + 1, M is
not a C6 subgroup. If M is C3 and C8 subgroups, a simple calculation shows that 2t−1 - |M |, a final
contradiction.
Theorem 3.1 is proved.
LetM be a class of groups. If there is no the section in a group G to be isomorphic to a member
of M, then G is called M-free. For the convenience, write = for the set of all PSL2(q), where
q = pf is odd and the order of Sylow 2-subgroup of PSL2(q) is 4.
Theorem 3.2. Let G be a group and N a normal subgroup of G, p ∈ π(G) and P ∈ Sylp(N).
Suppose that (|G|, p− 1) = 1 and G/N is p-nilpotent. If G is =-free and all maximal subgroups of
P are s-conditionally permutable in G, then G is p-nilpotent.
Proof. Assume that the result is false. Let (G,N) be a counterexample with |G|+ |N | minimal.
(1) G has a unique minimal normal subgroup L contained in N , G/L is p-nilpotent and L �
� Φ(G), and so L is not a p′-group.
Let L be a minimal normal subgroup ofG contained inN . Consider the quotient groupG = G/L.
Clearly G/N ∼= G/N is p-nilpotent and P = PL/L is a Sylow p-subgroup of N , where N = N/L.
Let P1 = P1L/L be a maximal subgroup of P . We may assume that P1 is a maximal subgroup
of P . By Lemma 2.1(1), P1 is s-conditionally permutable in G. The choice of G implies that G is
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862 Y. XU, X. H. LI
p-nilpotent. Since the class of p-nilpotent groups is a saturated formation, we may assume that L is
a unique minimal normal subgroup of G contained in N and L � Φ(G), and so L is not a p′-group.
(2) Op(N) = 1.
If not, then by (1), L ≤ Op(N) and, there is a maximal subgroup M of G such that G = LM
and L∩M = 1, so N = G∩N = L(M ∩N) and L∩ (M ∩N) = 1. It is clear that LMp ∈ Sylp(G)
and we may let (M ∩N)p < P , where (M ∩N)p ∈ Sylp(M ∩N). Let P1 be a maximal subgroup
of P containing (M ∩ N)p. Then P = P1L. By the hypothesis, P1 is a s-conditionally permutable
subgroup of G, then there exists a Sylow q-subgroup Q of G such that P1Q = QP1 for any q ∈ π(G),
where q 6= p. Let L1 = L ∩ P1. Then |L : L1| = |L : L ∩ P1| = |LP1 : P1| = |P : P1| = p. So L1 is
a maximal subgroup of L. If L ≤ P1Q, then P = LP1 ≤ P1Q, a contradiction. Hence L∩P1Q < L
and L1 = L ∩ P1Q. Consequently, L1 = L ∩ P1Q� P1Q, P1Q ≤ NG(L1). It is clear that L1 � L.
So P = LP1 ≤ NG(L1). By the arbitrariness of q ∈ π(G), we have L1 � G, hence L1 = 1 by
the minimal normality of L in G. This means that L is a cyclic subgroup of prime order. Since
G/CG(L) is isomorphic to a subgroup of Aut(L) and |Aut(L)| = p − 1, by (|G|, p − 1) = 1, we
have CG(L) = G, and L ≤ Z(G). Hence G = L ×M . Since M ∼= G/L, we get M is p-nilpotent
by (1), so G is p-nilpotent, a contradiction.
(3) End of the proof.
By (1) and (2), L is not solvable and so p = 2 by the Odd Order Theorem. Let L = T1×T2× . . .
. . .× Ts, where Ti are non-Abelian simple groups with Ti ∼= T1, 1 ≤ i ≤ s. Since P ∩ L ∈ Syl2(L),
we have P ∩ L = K1 × K2 × . . . × Ks, where Ki ∈ Syl2(Ti). Now we claim that there exists a
maximal subgroup P1 of P and i such that Ki ≤ P1. If P∩L < P , it is clear. Assume that P∩L = P .
Then (L,L) satisfies the hypothesis by Lemma 2.1(2). If L is a non-Abelian simple group, then every
maximal subgroup of P is s-conditionally permutable in L. By the hypothesis and Theorem 3.1, we
get L ∈ =, a contradiction. Hence L is not a non-Abelian simple group. Therefore, we can choose
the maximal subgroup P1 of P and i such that Ki ≤ P1. By the hypothesis, there exists a Sylow
q-subgroup Q of G such that P1Q = QP1 for any q ∈ π(G), where q 6= 2. Hence Ti ∩P1Q is a Hall
{2, q}-subgroup of Ti for any q ∈ π(T ) with q 6= 2. This contradicts the Lemma 2.2.
Theorem 3.2 is proved.
Corollary 3.1. Suppose that G is =-free. If for every prime p dividing the order of G and P ∈
∈ Sylp(G), every maximal subgroup of P is s-conditionally permutable in G, then G is a Sylow
tower group of supersolvable type.
Theorem 3.3. Let F be a saturated formation containing U , and G a group with a normal
subgroup N such that G/N ∈ F . If N is =-free and all maximal subgroups of every noncyclic Sylow
subgroup P of N are s-conditionally permutable in G, then G ∈ F .
Proof. Assume that the result is false and let (G,N) be a counterexample with |G| + |N |
minimal.
If all Sylow subgroups of N are cyclic, then all Sylow subgroups of F ∗(N) are cyclic. By
Lemma 2.3, G ∈ F . Therefore, when we want to prove G ∈ F in the following arguments, we
always assume that N has a noncyclic Sylow subgroup if (G,N) satisfies the hypothesis of (G,N)
in Theorem 3.3. By Lemma 2.1(2) and Corollary 3.1 N is a Sylow tower group of supersolvable type.
Let r be the largest prime in π(N) and R ∈ Sylr(N). Then R is normal in G and (G/R)/(N/R) ∼=
∼= G/N ∈ =. By Lemma 2.1(1), every maximal subgroup of any Sylow subgroup of N/R is s-
conditionally permutable in G/R. Therefore, G/R satisfies the hypotheses for the normal subgroup
N/R. Thus, by induction, G/R ∈ F , so R is noncyclic by Lemma 2.3. By Lemma 2.1(1), we may
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s-CONDITIONALLY PERMUTABLE SUBGROUPS AND p-NILPOTENCY OF FINITE GROUPS 863
assume that G has a unique minimal normal subgroup L which is contained in R and G/L ∈ F . If
L ≤ Φ(G), then it follows thatG ∈ F , a contradiction. Thus, we may further assume that R∩Φ(G) =
= 1. Then, by Lemma 2.4, R = F (R) = L is an elementary abelian minimal normal subgroup of G.
Since R = L � Φ(G), we may choose a maximal subgroup M of G such that R �M . Let Mr be a
Sylow r-subgroup of M . Then G = RM , R ∩M = 1 and Gr = RMr is a Sylow r-subgroup of G.
Let G1 be a maximal subgroup of Gr containing Mr. Then R ∩G1 is a maximal subgroup of R. By
the hypothesis, R ∩G1 is s-conditionally permutable in G, so there exists a Q ∈ Sylq(G) such that
(R∩G1)Q = Q(R∩G1) with q 6= r, thus R∩G1 = (R∩G1)(R∩Q) = R∩(R∩G1)Q�(R∩G1)Q,
hence (R ∩G1)Q ≤ NG(R ∩G1). Clearly, R ∩G1 �Gr. Therefore, R ∩G1 �G. By the minimal
normality of R in G, we have R ∩G1 = 1. Hence |R| = r, R is cyclic, a contradiction.
Theorem 3.3 is proved.
Corollary 3.2 ([10], Theorem 4.2). Let F be a saturated formation containing U , and G a group
with a solvable normal subgroup N such that G/N ∈ F . If all maximal subgroups of every noncyclic
Sylow subgroup P of N are s-conditionally permutable in G, then G ∈ F .
1. Aschbacher M. On the maximal subgroups of the finite classical groups // Invent. Math. – 1984. – 76. – P. 465 – 514.
2. Xianhua Li. Influence of indices of the maximal subgroups of the finite simple groups on the structure of a finite
group // Communs Algebra. – 2004. – 32, № 1. – P. 33 – 64.
3. Jianxing Bi, Xianhua Li. A characterization of alternating groups by orders of solvable subgroups // J. Algebra and
Appl. – 2004. – 3, № 4. – P. 445 – 452.
4. Cohen A. M., Liebeck M. W., Jan Saxl, Gary M. Seitz. The local maximal subgroups of exceptional groups of Lie-type,
finite and algebaic // Proc. London Math. Soc. – 1992. – 64, № 3. – P. 21 – 48.
5. Conway J. H., Curtis R. T., Norton S. P., Parker R. A., Wilson R. A. Atlas of finite groups. – Oxford: Clarendon Press,
1985.
6. Cooperstein B. N. The maximal subgroups of G2(2n) // J. Algebra. – 1981. – 70. – P. 23 – 36.
7. Delandtsheer A. Flage-transitive finite simple groups // Arch. Math. – 1986. – 47. – P. 395 – 400.
8. Feit W. On large Zsigmondy primes // Proc. Amer. Math. Soc. – 1988. – 102, № 1. – P. 29 – 33.
9. Guralnick R. M., Penttila T., Praeger C. E., Saxl J. Linear groups with orders having certain large prime divisors //
Proc. London Math. Soc. – 1999. – 78, № 3. – P. 167 – 214.
10. Jianhong Huang, Wenbin Guo. s-Conditionally permutable subgroups of finite groups // Ann. Math. China A. –
2007. – 28, № 1. – P. 17 – 26.
11. Huppert B. Endliche Gruppen I. – New York; Berlin: Springer, 1967.
12. Kimmerle W., Lyons R., Sandling R., Teague D. N. Composition factors from the group ring and Artin’s theorem on
orders of simple groups // Proc. London Math. Soc. – 1990. – 60, № 3. – P. 89 – 122.
13. Kleidman P., Liebeck M. The subgroup structure of the finite classical groups. – Cambridge: Cambridge Univ. Press,
1990.
14. Kleidman P. The low-dimensional finite classical groups and their subgroups. – Preprint, 1988.
15. Kleidman P. B. The maximal subgroups of the finite 8-dimensional orthogonal groups PΩ+
8 (q) and of their automor-
phism groups // J. Algebra. – 1987. – 110. – P. 173 – 242.
16. Kleidman P. B. The maximal subgroups of the Chevalley groups G2(q) with q odd, the Ree groups 2G2(q), and of
their automorphism groups // J. Algebra. – 1988. – 117. – P. 30 – 71.
17. Kleidman P. B. The maximal subgroups of the Steinberg triality groups 3D4(q) and of their automorphism groups //
J. Algebra. – 1988. – 115. – P. 182 – 199.
18. Liebeck M. W. On the order of maximal subgroup of the finite classical groups // Proc. London Math. Soc. – 1985. –
50, № 3. – P. 426 – 446.
19. Liebeck M. W., Prager C. E., Saxl J. The maximal factorization of the finite simple groups and their automorphism
groups // Mem. Amer. Math. Soc. – 1990. – 86(432). – P. 1 – 151.
20. Liebeck M. W., Saxl J. The primitive permutation groups of odd degree // J. London Math. Soc. – 1985. – 31, № 2. –
P. 250 – 264.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6
864 Y. XU, X. H. LI
21. Liebeck M. W., Saxl J. On the order of maximal subgroup of the finite exceptional groups of Lie-type // Proc. London
Math. Soc. – 1987. – 55, № 3. – P. 299 – 330.
22. Liebeck M. W., Seitz G. M. A survey of maximal subgroups of exceptional groups of Lie-type // Groups, Combinatories
& Geometry. – River Edge, NJ: World Sci. Publ., 2003. – P. 139 – 146.
23. Gunter Malle. The maximal subgroups of 2F4(q2) // J. Algebra. – 1991. – 139, № 1. – P. 52 – 69.
24. Yangming Li, Xianhua Li. Z-permutable subgroups and p-nilpotency of finite groups // J. Pure and Appl. Algebra. –
2005. – 202. – P. 72 – 81.
25. Ramadan M. Influence of normality on maximal subgroups of Sylow subgroups of a finite group // Acta Math.
hung. – 1992. – 59, № 1-2. – P. 107 – 110.
26. Skiba A. N. A note on c-normal subgroups of finite groups // Algebra and Discrete Math. – 2005. – 3. – P. 85 – 95.
27. Skiba A. N. On weakly s-permutable subgroups of finite groups // J. Algebra. – 2007. – 315, № 1. – P. 192 – 209.
28. Srinivasan S. Two sufficient conditions for supersolvability of finite groups // Isr. J. Math. – 1980. – 35, № 3. –
P. 210 – 214.
29. Xu Y., Zhao T., Li X. H. On CISE-normal subgroups of finite groups // Turish J. Math. – 2012. – 36, № 2. –
P. 231 – 264.
Received 17.07.12
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| id | umjimathkievua-article-2184 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:20:18Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c9/2d23e7275dbf6f64e7d216db37a02ac9.pdf |
| spelling | umjimathkievua-article-21842019-12-05T10:25:45Z $s$-Conditionally Permutable Subgroups and $p$-Nilpotency of Finite Groups $s$-Умовно переставні підгрупи та $p$-нільпотентність скінченних груп Li, X. H. Xu, Y. Лі, Х. Х. Ху, Я. We study the $p$-nilpotency of a group such that every maximal subgroup of its Sylow $p$-subgroups is $s$-conditionally permutable for some prime $p$. By using the classification of finite simple groups, we get interesting new results and generalize some earlier results. Вивчено $p$-нільпотентність групи, для якої кожна максимальна підгрупа її силовських $p$-підгруп є $s$-умовно переставною для деякого простого $p$. За допомогою класифiкацiї скінченних простих груп отримано цікаві нові результати та узагальнено деякі результати, що отримані раніше. Institute of Mathematics, NAS of Ukraine 2014-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2184 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 6 (2014); 858–864 Український математичний журнал; Том 66 № 6 (2014); 858–864 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2184/1369 https://umj.imath.kiev.ua/index.php/umj/article/view/2184/1370 Copyright (c) 2014 Li X. H.; Xu Y. |
| spellingShingle | Li, X. H. Xu, Y. Лі, Х. Х. Ху, Я. $s$-Conditionally Permutable Subgroups and $p$-Nilpotency of Finite Groups |
| title | $s$-Conditionally Permutable Subgroups and $p$-Nilpotency of Finite Groups |
| title_alt | $s$-Умовно переставні підгрупи та $p$-нільпотентність скінченних груп |
| title_full | $s$-Conditionally Permutable Subgroups and $p$-Nilpotency of Finite Groups |
| title_fullStr | $s$-Conditionally Permutable Subgroups and $p$-Nilpotency of Finite Groups |
| title_full_unstemmed | $s$-Conditionally Permutable Subgroups and $p$-Nilpotency of Finite Groups |
| title_short | $s$-Conditionally Permutable Subgroups and $p$-Nilpotency of Finite Groups |
| title_sort | $s$-conditionally permutable subgroups and $p$-nilpotency of finite groups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2184 |
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