On weakly s -normal subgroups of finite groups
Assume that G is a finite group and H is a subgroup of G. We say that H is s-permutably imbedded in G if, for every prime number p that divides |H|, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-permutable subgroup of G; a subgroup H is s-semipermutable in G if HGp=GpH for any Sylow p...
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irk-123456789-1663922020-02-20T01:27:13Z On weakly s -normal subgroups of finite groups Li, Yangming Qiao, Shouhong Статті Assume that G is a finite group and H is a subgroup of G. We say that H is s-permutably imbedded in G if, for every prime number p that divides |H|, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-permutable subgroup of G; a subgroup H is s-semipermutable in G if HGp=GpH for any Sylow p-subgroup Gp of G with (p,|H|)=1; a subgroup H is weakly s-normal in G if there are a subnormal subgroup T of G and a subgroup H∗ of H such that G=HT and H⋂T≤H∗, where H∗ is a subgroup of H that is either s-permutably imbedded or s-semipermutable in G. We investigate the influence of weakly s-normal subgroups on the structure of finite groups. Some recent results are generalized and unified. Нехай G — скiнченна група, а H — пiдгрупа G. Будемо говорити, що H є s-переставно вкладеною в G, якщо для будь-якого простого числа p, що дiлить |H|, силовська p-пiдгрупа H є також силовською p-пiдгрупою деякої s-переставної пiдгрупи G; H є s-напiвпереставною в G, якщо HGp=GpH для будь-якої силовської p-пiдгрупи Gp групи G iз (p,|H|)=1; H є слабко s-нормальною в G, якщо iснують субнормальна пiдгрупа T групи G i пiдгрупа H∗ пiдгрупи H такi, що G=HT i H⋂T≤H∗, де H∗ — пiдгрупа H, що є або s-переставно вкладеною, або s-напiвпереставною в G. Дослiджено вплив слабко s-нормальних пiдгруп на будову скiнченних груп. Узагальнено та унiфiковано деякi нещодавнi результати. 2011 Article On weakly s -normal subgroups of finite groups / Yangming Li, Shouhong Qiao // Український математичний журнал. — 2011. — Т. 63, № 11. — С. 1555–1564. — Бібліогр.: 23 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166392 512.5 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Li, Yangming Qiao, Shouhong On weakly s -normal subgroups of finite groups Український математичний журнал |
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Assume that G is a finite group and H is a subgroup of G. We say that H is s-permutably imbedded in G if, for every prime number p that divides |H|, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-permutable subgroup of G; a subgroup H is s-semipermutable in G if HGp=GpH for any Sylow p-subgroup Gp of G with (p,|H|)=1; a subgroup H is weakly s-normal in G if there are a subnormal subgroup T of G and a subgroup H∗ of H such that G=HT and H⋂T≤H∗, where H∗ is a subgroup of H that is either s-permutably imbedded or s-semipermutable in G. We investigate the influence of weakly s-normal subgroups on the structure of finite groups. Some recent results are generalized and unified. |
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| author |
Li, Yangming Qiao, Shouhong |
| author_facet |
Li, Yangming Qiao, Shouhong |
| author_sort |
Li, Yangming |
| title |
On weakly s -normal subgroups of finite groups |
| title_short |
On weakly s -normal subgroups of finite groups |
| title_full |
On weakly s -normal subgroups of finite groups |
| title_fullStr |
On weakly s -normal subgroups of finite groups |
| title_full_unstemmed |
On weakly s -normal subgroups of finite groups |
| title_sort |
on weakly s -normal subgroups of finite groups |
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Інститут математики НАН України |
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2011 |
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Статті |
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| citation_txt |
On weakly s -normal subgroups of finite groups / Yangming Li, Shouhong Qiao // Український математичний журнал. — 2011. — Т. 63, № 11. — С. 1555–1564. — Бібліогр.: 23 назв. — англ. |
| series |
Український математичний журнал |
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AT liyangming onweaklysnormalsubgroupsoffinitegroups AT qiaoshouhong onweaklysnormalsubgroupsoffinitegroups |
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2025-07-14T21:22:11Z |
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2025-07-14T21:22:11Z |
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1837658943535448064 |
| fulltext |
UDC 512.5
Yangming Li (Guangdong Univ. Education, China),
Shouhong Qiao (Yunnan Univ., Kunming, China)
ON WEAKLY s-NORMAL SUBGROUPS OF FINITE GROUPS*
ПРО СЛАБКО s-НОРМАЛЬНI ПIДГРУПИ
СКIНЧЕННИХ ГРУП
Assume that G is a finite group and H is a subgroup of G. We say that H is s-permutably imbedded in
G if, for every prime number p that divides |H|, a Sylow p-subgroup of H is also a Sylow p-subgroup of
some s-permutable subgroup of G; a subgroup H is s-semipermutable in G if HGp = GpH for any Sylow
p-subgroup Gp of G with (p, |H|) = 1; a subgroup H is weakly s-normal in G if there are a subnormal
subgroup T of G and a subgroup H∗ of H such that G = HT and H ∩ T ≤ H∗, where H∗ is a subgroup
of H that is either s-permutably imbedded or s-semipermutable in G. We investigate the influence of weakly
s-normal subgroups on the structure of finite groups. Some recent results are generalized and unified.
Нехай G — скiнченна група, а H — пiдгрупа G. Будемо говорити, що H є s-переставно вкладеною в
G, якщо для будь-якого простого числа p, що дiлить |H|, силовська p-пiдгрупа H є також силовською
p-пiдгрупою деякої s-переставної пiдгрупи G; H є s-напiвпереставною в G, якщо HGp = GpH для
будь-якої силовської p-пiдгрупи Gp групи G iз (p, |H|) = 1; H є слабко s-нормальною в G, якщо
iснують субнормальна пiдгрупа T групи G i пiдгрупа H∗ пiдгрупи H такi, що G = HT i H ∩T ≤ H∗,
деH∗ — пiдгрупаH, що є або s-переставно вкладеною, або s-напiвпереставною вG. Дослiджено вплив
слабко s-нормальних пiдгруп на будову скiнченних груп. Узагальнено та унiфiковано деякi нещодавнi
результати.
1. Introduction. All groups considered in this paper will be finite. We use conventional
notions and notation, as in Huppert [1]. G always denotes a group, |G| is the order of
G, π(G) denotes the set of all primes dividing |G|, Gp is a Sylow p-subgroup of G for
some p ∈ π(G).
Let F be a class of groups. We call F a formation provided that (i) if G ∈ F and
H/G, then G/H ∈ F , and (ii) if G/M and G/N are in F , then G/(M∩N) is in F for
all normal subgroups M,N of G. A formation F is said to be saturated if G/Φ(G) ∈ F
implies that G ∈ F . In this paper, U will denote the class of all supersolvable groups.
Clearly, U is a saturated formation (ref. [1, p. 713], Satz 8.6).
Two subgroups H and K of G are said to be permutable if HK = KH. A subgroup
H of G is said to be s-permutable (or s-quasinormal, π-quasinormal) [2] in G if H
permutes with every Sylow subgroup of G; H is said c-normal [3] in G if G has a
normal subgroup T such that G = HT and H ∩T ≤ HG, where HG is the normal core
of H in G. More recently, Skiba in [4] introduces the following concept, which covers
both s-permutability and c-normality:
Definition 1.1. Let H be a subgroup of G. H is called weakly s-permutable in
G if there is a subnormal subgroup T of G such that G = HT and H ∩ T ≤ HsG,
where HsG is the subgroup of H generated by all those subgroups of H which are
s-permutable in G.
From [5], we know that a subgroupH ofG is said to be s-permutably embedded inG
if for each prime p dividing |H|, a Sylow p-subgroup of H is also a Sylow p-subgroup
of some s-permutable subgroup of G. In [6], we give a new concept which covers
properly both s-permutably embedding property and Skiba’s weakly s-permutability.
*Project supported in part by NSFC(11171353/A010201) and NSF of Guangdong Province
(S2011010004447).
c© YANGMING LI, SHOUHONG QIAO, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11 1555
1556 YANGMING LI, SHOUHONG QIAO
Definition 1.2. Let H be a subgroup of G. We say that H is weakly s-permutably
embedded in G if there are a subnormal subgroup T of G and an s-permutably embed-
ded subgroup Hse of G contained in H such that G = HT and H ∩ T ≤ Hse.
In another direction, a subgroup H of G is said to be s-semipermutable [7] in G
if H permutes with every Sylow p-subgroup Gp of G with (|H|, p) = 1. It is easy to
give concrete examples to show that s-semipermutablity and s-permutably embedding
property are not equivalent. Here, we introduce a new concept which covers properly
both s-semipermutability and weakly s-permutably embedding property.
Definition 1.3. Let H be a subgroup of G. We say that H is weakly s-normal in G
if there are a subnormal subgroup T of G and a subgroup H∗ of H such that G = HT
and H ∩T ≤ H∗, where H∗ is a subgroup of H which is either s-permutably embedded
or s-semipermutable in G.
Remark. Obviously, weakly s-permutably embedding property (or s-semipermut-
ability) implies weakly s-normality by the definitions. The converse does not hold in
general.
Examples. 1. Suppose that G = A5, the alternative group of degree 5. Then A4 is
weakly s-normal in G, but not weakly s-permutably embedded in G.
2. Suppose that G = S4, the symmetric group of degree 4. Take H = 〈(34)〉. Then
H is weakly s-normal in G, but not s-semipermutable in G.
In the literature, authors usually put the assumptions on either the minimal subgroups
(and cyclic subgroups of order 4 when p = 2) or the maximal subgroups of some kinds
of subgroups of G when investigating the structure of G, such as in [7 – 13, 16 – 21]
etc. In the nice paper [4], Skiba provided a unified viewpoint for a series of similar
problems.
For the sake of convenience of statement, we introduce the following notation.
Let P be a p-subgroup of G for some p ∈ π(G). We say that P satisfies (∗) ((∗)′,
(4), (♦1), (♦2), (♦3), (♦4), respectively) in G if
(∗): P has a subgroup D such that 1 < |D| < |P | and all subgroups H of P
with order |H| = |D| and with order |H| = 2|D| (if P is a non-abelian 2-group and
|P : D| > 2) are weakly s-permutable in G.
(∗)′: P has a subgroup D such that 1 < |D| < |P | and all subgroups H of P
with order |H| = |D| are weakly s-permutable in G. When P is a non-abelian 2-group
and |P : D| > 2, in addition, the subgroup H of P is weakly s-permutable in G if
|H| = 2|D| and exp (H) > 2.
(4): P has a subgroup D such that 1 < |D| < |P | and all subgroups H of P with
order |H| = |D| are weakly s-permutably embedded in G. When p = 2 and |P : D| > 2,
in addition, the subgroup H of P is weakly s-permutably embedded in G if |H| = 2|D|
and exp (H) > 2.
(♦1): P has a subgroup D such that 1 < |D| < |P | and all subgroups H of P
with order |H| = |D| are weakly s-normal in G. When P is a non-abelian 2-group and
|P : D| > 2, in addition, H is weakly s-normal in G if |H| = 2|D| and exp (H) > 2.
(♦2): P has a subgroup D such that 1 < |D| < |P | and all subgroups H of P with
order |H| = |D| are either s-permutably embedded or s-semipermutable in G. When P
is a non-abelian 2-group and |P : D| > 2, in addition, the subgroup H of P is either
s-permutably embedded or s-semipermutable in G if |H| = 2|D| and exp (H) > 2.
(♦3): P has a subgroup D such that 1 < |D| < |P | and all subgroups H of P
with order |H| = |D| are s-semipermutable in G. When P is a non-abelian 2-group and
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11
ON WEAKLY s-NORMAL SUBGROUPS OF FINITE GROUPS 1557
|P : D| > 2, in addition, the subgroup H of P is s-semipermutable in G if |H| = 2|D|
and exp (H) > 2.
(♦4): P has a subgroup D such that 1 < |D| < |P | and all subgroups H of P with
order |H| = |D| are either s-semipermutable or c-normal in G. When P is a non-abelian
2-group and |P : D| > 2, in addition, the subgroup H of P is either s-semipermutable
or c-normal in G if |H| = 2|D| and exp (H) > 2.
The following is the main result of [4].
Theorem 1.1 ([4], Theorem 1.3). Let F be a saturated formation containing U and
G a group with E as a normal subgroup of G such that G/E ∈ F . Suppose that every
non-cyclic Sylow subgroup P of F ∗(E) satisfies (∗) in G. Then G ∈ F .
Scrutinizing the proof of [4] (Theorem 1.3), we can find that the following theorem
holds:
Theorem 1.2. Let F be a saturated formation containing U and G a group with a
normal subgroup E such that G/E ∈ F . Suppose that every non-cyclic Sylow subgroup
P of F ∗(E) satisfies (∗)′ in G. Then G ∈ F .
In [6], Theorem 1.2 was extended as follows.
Theorem 1.3. Let F be a saturated formation containing U and G a group with
E as a normal subgroup of G such that G/E ∈ F . If every non-cyclic Sylow subgroup
of F ∗(E) satisfies 4 in G, then G ∈ F .
In [22], there holds the following result.
Theorem 1.4. Let F be a saturated formation containing U and G a group with a
normal subgroup E such that G/E ∈ F . If every non-cyclic Sylow subgroup of F ∗(E)
satisfies ♦3 in G, then G ∈ F .
In [23], Theorem 1.4 was extended as follows.
Theorem 1.5. Let F be a saturated formation containing U and G a group with a
normal subgroup E such that G/E ∈ F . If every non-cyclic Sylow subgroup of F ∗(E)
satisfies ♦4 in G, then G ∈ F .
In this paper, the main purpose is to generalize results mentioned above as Theo-
rem 3.4. Theorem 3.2 related to p-nilpotency of groups is a main step in the proof of
Theorem 3.4.
2. Preliminaries.
Lemma 2.1. Suppose that H is an s-semipermutable subgroup of G. Then
(a) If H ≤ K ≤ G, then H is s-semipermutable in K.
(b) Let N be a normal subgroup of G. If H is a p-group for some prime p ∈ π(G),
then HN/N is s-semipermutable in G/N.
(c) If H ≤ Op(G), then H is s-permutable in G.
Proof. By [7].
Lemma 2.2 ([5], Lemma 1). Suppose that U is s-permutably embedded in a group
G, and that H ≤ G and N �G.
(a) If U ≤ H, then U is s-permutably embedded in H.
(b) UN is s-permutably embedded in G and UN/N is s-permutably embedded in
G/N.
Lemma 2.3 ([21], Lemma 2.3). Suppose that H is s-permutable in G, P a Sylow
p-subgroup of H, where p is a prime. If HG = 1, then P is s-permutable in G.
Lemma 2.4 ([21], Lemma 2.4). Suppose P is a p-subgroup ofG contained inOp(G).
If P is s-permutably embedded in G, then P is s-permutable in G.
Now we give some basic properties of weakly s-normality.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11
1558 YANGMING LI, SHOUHONG QIAO
Lemma 2.5. Let U be a weakly s-normal subgroup of G and N a normal sub-
group of G. Then
(a) If U ≤ H ≤ G, then U is weakly s-normal in H.
(b) Suppose that U is a p-group for some prime p. If N ≤ U, then U/N is weakly
s-normal in G/N.
(c) Suppose that U is a p-group for some prime p and N is a p′-subgroup. Then
UN/N is weakly s-normal in G/N.
(d) Suppose that U is a p-group for some prime p and U is neither s-semipermutable
nor s-permutably embedded inG. ThenG has a normal subgroupM such that |G : M | =
= p and G = MU.
(e) If U ≤ Op(G) for some prime p, then U is weakly s-permutable in G.
Proof. By the hypotheses, there are a subnormal subgroup T of G and a subgroup
U∗ of U such that G = UT and U ∩ T ≤ U∗, where U∗ is a subgroup of U which is
either s-permutably embedded or s-semipermutable in G.
(a) H = U(H∩T ). Obviously H∩T is subnormal in H and U∩(H∩T ) = U∩T ≤
≤ U∗. By Lemmas 2.1 and 2.2, we know that U∗ is either s-permutably embedded or
s-semipermutable in H. Hence U is weakly s-normal in H.
(b) G/N = (U/N)(TN/N). Obviously TN/N is subnormal in G/N and (U/N)∩
∩ (TN/N) = (U ∩ TN)/N = (U ∩ T )N/N ≤ U∗N/N. By Lemmas 2.1 and 2.2,
we know that U∗N/N is either s-permutably embedded or s-semipermutable in G/N.
Hence U/N is weakly s-normal in G/N.
(c) It is easy to see that N ≤ T and G/N = (UN/N)(T/N). We have T/N is
subnormal in G/N and (UN/N)∩ (T/N) = (U ∩T )N/N ≤ U∗N/N. By Lemmas 2.1
and 2.2, we know that U∗N/N is either s-permutably embedded or s-semipermutable
in G/N. Hence U/N is weakly s-normal in G/N.
(d) If T = G, then U = U ∩ T ≤ U∗ ≤ U. Thus U = U∗ is either s-semipermutable
or s-permutably embedded in G, contrary to the hypotheses. Consequently, T is a proper
subgroup of G. Hence G has a proper normal subgroup K such that T ≤ K. Since G/K
is a p-group, G has a normal maximal subgroup M such that |G : M | = p and G = MU.
(e) By Lemmas 2.1(c) and 2.4 and the definitions.
Lemma 2.6 ([14], A, 1.2). Let U, V and W be subgroups of a group G. Then the
following statements are equivalent.
(a) U ∩ VW = (U ∩ V )(U ∩W ).
(b) UV ∩ UW = U(V ∩W ).
Lemma 2.7 ([1], VI, 4.10). Assume that A and B are two subgroups of a group G
and G 6= AB. If ABg = BgA holds for any g ∈ G, then either A or B is contained in
a proper normal subgroup of G.
Lemma 2.8 ([1], III, 5.2 and IV, 5.4). Suppose that p is a prime and G is a minimal
non-p-nilpotent group, i.e., G is not a p-nilpotent group but whose proper subgroups are
all p-nilpotent. Then
(a) G has a normal Sylow p-subgroup P and G = PQ, where Q is a non-normal
cyclic q-subgroup of G for some prime q 6= p.
(b) P/Φ(P ) is a minimal normal subgroup of G/Φ(P ).
(c) The exponent of P is p or 4.
The generalized Fitting subgroup F ∗(G) of G is the unique maximal normal quasi-
nilpotent subgroup of G. Its definition and important properties can be found in [15] (X,
13). We would like to give the following basic facts we will use in our proof.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11
ON WEAKLY s-NORMAL SUBGROUPS OF FINITE GROUPS 1559
Lemma 2.9 ([15], X, 13). Let G be a group and M a subgroup of G.
(a) If M is normal in G, then F ∗(M) ≤ F ∗(G).
(b) F ∗(G) 6= 1 if G 6= 1; in fact, F ∗(G)/F (G) = soc (F (G)CG(F (G))/F (G)).
(c) F ∗(F ∗(G)) = F ∗(G) ≥ F (G); if F ∗(G) is solvable, then F ∗(G) = F (G).
3. Main results.
Theorem 3.1. Let G be a group and P = Gp a Sylow p-subgroup of G, where p
is the smallest prime dividing |G|. If all maximal subgroups of P are weakly s-normal
in G, then G is p-nilpotent.
Proof. Suppose that the theorem is false and G is a counter-example with minimal
order. We will derive a contradiction in several steps.
Step 1. G has a unique minimal normal subgroup N such that G/N is p-nilpotent
and Φ(G) = 1.
Let N be a minimal normal subgroup of G. Consider G/N, we will show that G/N
satisfies the hypotheses of the theorem. Let M/N be a maximal subgroup of PN/N. It
is easy to see M = P1N for some maximal subgroup P1 of P. It follows that P ∩N =
= P1∩N is a Sylow subgroup of N. By the hypotheses, there are a subnormal subgroup
K1 of G and a subgroup (P1)∗ of P1 such that G = P1K1 and P1∩K1 ≤ (P1)∗, where
(P1)∗ is a subgroup of P1 which is either s-permutably embedded or s-semipermutable
in G. Then G/N = M/N ·K1N/N = P1N/N ·K1N/N. It is easy to see that K1N/N
is subnormal in G/N. Since (|N : P1 ∩N |, |N : K1 ∩N |) = 1, (P1 ∩N)(K1 ∩N) =
= N = N∩G = N∩(P1K1). By Lemma 2.6, (P1N)∩(K1N) = (P1∩K1)N. It follows
from Lemmas 2.1 and 2.2 that (P1N/N)∩ (K1N/N) = (P1 ∩K1)N/N ≤ (P1)∗N/N,
(P1)∗N/N is either s-permutably embedded or s-semipermutable in G/N. Hence M/N
is weakly s-normal in G/N. Therefore G/N satisfies the hypotheses of the theorem. The
choice of G yields that G/N is p-nilpotent. The uniqueness of N and Φ(G) = 1 are
obvious.
Step 2. Op′(G) = 1.
If Op′(G) 6= 1, then N ≤ Op′(G) by Step 1. By Lemma 2.5(c), G/N satisfies
the hypotheses, hence G/N is p-nilpotent. Now the p-nilpotency of G/N implies the
p-nilpotency of G, a contradiction.
Step 3. Op(G) = 1 and G = PN. Therefore G is not solvable and N is a direct
product of isomorphic non-abelian simple groups.
If Op(G) 6= 1, Step 1 yields N ≤ Op(G) and Φ(Op(G)) ≤ Φ(G) = 1. Therefore G
has a maximal subgroup M such that G = MN and M ∩ N = 1. Since Op(G) ∩M
is normalized by N and M, Op(G) ∩M is normal in G. The uniqueness of N yields
N = Op(G). Clearly P = N(P ∩M). Since P ∩M < P, there exists a maximal
subgroup P1 of P such that P ∩M ≤ P1. Then P = NP1. By the hypotheses, there
are a subnormal subgroup T of G and a subgroup (P1)∗ of P1 such that G = P1T
and P1 ∩ T ≤ (P1)∗, where (P1)∗ is a subgroup of P1 which is either s-permutably
embedded or s-semipermutable in G. Since N ≤ Op(G) ≤ T by Step 1, we have
P1 ∩N = (P1)∗ ∩N.
If (P1)∗ is s-semipermutable in G, then, for any Sylow q-subgroup Gq of G, q 6= p,
there holds
[P1 ∩N,Gq] ≤ N ∩ (P1)∗Gq = N ∩ (P1)∗ = N ∩ P1.
Obviously, P1∩N is normalized by P. Therefore P1∩N is normal in G. The minimality
of N implies that P1 ∩ N = 1. Hence N is of order p. Thus G is p-nilpotent, a
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1560 YANGMING LI, SHOUHONG QIAO
contradiction. Hence P1 is s-permutably embedded in G. Then we get a contradiction
with the same argument in the Step 3 of the proof of [6] (Theorem 3.1).
If PN < G, then PN is p-nilpotent. Hence N is p-nilpotent. Therefore N = Np ≤
≤ Op(G) = 1 by Step 2, a contradiction. Hence G = PN.
By Step 2, we can see that G is not solvable and N is a direct product of isomorphic
non-abelian simple groups. Thus Step 3 holds.
Step 4. The final contradiction.
If N ∩ P ≤ Φ(P ), then N is p-nilpotent by Tate’s theorem [1, p. 431] (Satz 4.7),
contrary to Step 3. Consequently, there is a maximal subgroup P1 of P such that P =
= (N ∩ P )P1. Since P1 is weakly s-normal in G, by the hypotheses, there are a
subnormal subgroup T of G and a subgroup (P1)∗ of P1 such that G = P1T and
P1∩T ≤ (P1)∗, where (P1)∗ is a subgroup of P1 which is either s-permutably embedded
or s-semipermutable in G.
Suppose that (P1)∗ is s-semipermutable in G. Since G = PN, any Sylow q-
subgroup Nq of N is a Sylow q-subgroup of G, where q 6= p. We have (P1)∗Nq ≤ G,
and thus (P1)∗Nq ∩ N is a proper subgroup of N since N is nonsolvable. Then
N ∩ (P1)∗Nq = ((P1)∗ ∩ N)Nq < N . Applying Lemma 2.7, we know that N has
a proper normal subgroup M such that either (P1)∗ ∩ N ≤ M or Nq ≤ M. Since M
is proper in N, by [1] (I, Satz 9.12(b)), M contains no Sylow subgroups of N. Thus
(P1)∗ ∩N ≤M. Noticing that P1 ∩N = (P1)∗ ∩N ≤ P1 ∩M, we have
|N/M |p =
|N |p
|M |p
= |P ∩N : P ∩M | ≤ |P ∩N : P1 ∩N | ≤ |P : P1| = p.
Hence N/M is p-nilpotent by [1] (IV, Satz 2.8), but this is a contradiction.
Hence P1 is s-permutably embedded in G. Now we get the final contradiction with
the same argument in the Step 4 of the proof of [6] (Theorem 3.1).
This completes the proof of Theorem 3.1.
Theorem 3.2. Let G be a group and P a Sylow p-subgroup of G, where p is the
smallest prime dividing |G|. If P satisfies ♦1 in G, then G is p-nilpotent.
Proof. Suppose that the theorem is false and G is a counter-example with minimal
order. We will derive a contradiction in several steps.
Step 1. Op′(G) = 1.
Assume that Op′(G) 6= 1. Lemma 2.5(c) guarantees that G/Op′(G) satisfies the
hypotheses of the theorem. Thus G/Op′(G) is p-nilpotent by the choice of G. Then G
is p-nilpotent, a contradiction.
Step 2. |P : D| > p.
By Theorem 3.1.
Step 3. G has no subgroup of index p.
Suppose that G has a subgroup M such that |G : M | = p. Then M �G. By Step 2
together with induction,M is p-nilpotent, consequently,G is p-nilpotent, a contradiction.
Step 4. |D| > p.
Assume that |D| = p. Since G is not p-nilpotent, G has a minimal non-p-nilpotent
subgroup G1. By Lemma 2.8(a), G1 = [P1]Q, where P1 ∈ Syl p(G1) and Q ∈
∈ Sylq(G1), p 6= q. Denote Φ = Φ(P1). Let X/Φ be a subgroup of P1/Φ of order
p, x ∈ X \Φ and L = 〈x〉. Then L is of order p or 4 by Lemma 2.8(c). By the hypothe-
ses, L is weakly s-normal in G, thus in G1 by Lemma 2.5(a). Since L ≤ P1 = Op(G1),
by Lemma 2.5(e), L is weakly s-permutable in G1. Since G1 is a minimal non-p-
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ON WEAKLY s-NORMAL SUBGROUPS OF FINITE GROUPS 1561
nilpotent subgroup, G1 has no subgroup of index p. Thus, by [4] (Lemma 2.10(5)), L is
s-permutable in G1. Then X/Φ = LΦ/Φ is s-permutable in G1/Φ. [4] (Lemma 2.11)
implies that |P1/Φ| = p since P1/Φ is minimal normal in G1/Φ. It follows immediately
that P1 is cyclic. Hence G1 is p-nilpotent by [1] (Lemma 2.11), contrary to the choice
of G1.
Step 5. P satisfies ♦2 in G.
Assume that H ≤ P such that |H| = |D| and H is neither s-permutably embedded
nor s-semipermutable in G. By Lemma 2.5(d), there is a normal subgroup M of G such
that |G : M | = p, contrary to Step 3.
Step 6. If N is minimal normal in G contained in P, then |N | ≤ |D|.
Suppose that |N | > |D|. Since N ≤ Op(G), N is elementary abelian. By Lemma
2.5(e) and [4] (Lemma 2.11), N has a maximal subgroup which is normal in G, contrary
to the minimality of N.
Step 7. If N is minimal normal in G contained in P, then G/N is p-nilpotent.
If |N | < |D|, G/N satisfies the hypotheses of the theorem by Lemmas 2.1(b) and
2.2. Thus G/N is p-nilpotent by the minimal choice of G. So we may suppose that
|N | = |D| by Step 6. We will show that every cyclic subgroup of P/N of order p
or order 4 (when P/N is a non-abelian 2-group) is either s-permutably embedded or
s-semipermutable in G/N. Let K ≤ P with |K/N | = p. By Step 4, N is non-cyclic, so
are all subgroups containing N. Hence there is a maximal subgroup L 6= N of K such
that K = NL. Of course, |N | = |D| = |L|. Since L is either s-permutably embedded
or s-semipermutable in G by the hypotheses and Step 5, K/N = LN/N is either s-
permutably embedded or s-semipermutable in G/N by Lemmas 2.1(b) and 2.2. If p = 2
and P/N is non-abelian, take a cyclic subgroup X/N of P/N of order 4. Let K/N be
maximal in X/N. Then K is maximal in X and |K/N | = 2. Since X is non-cyclic
and X/N is cyclic, there is a maximal subgroup L of X such that N is not contained
in L. Thus X = LN and |L| = |K| = 2|D|. Since X/N = LN/N ∼= L/(L ∩ N) is
cyclic of order 4, by the hypotheses and Step 5, L is either s-permutably embedded or
s-semipermutable in G. By Lemmas 2.1 and 2.2, X/N = LN/N is either s-permutably
embedded or s-semipermutable in G/N. Hence P/N satisfies ♦2 in G/N. By the
minimal choice of G, G/N is p-nilpotent.
Step 8. Op(G) = 1.
Suppose that Op(G) 6= 1. Take a minimal normal subgroup N of G contained in
Op(G). By Step 7, G/N is p-nilpotent. This means that G has a subgroup of index p,
contrary to Step 3.
Step 9. Each minimal normal subgroup of G is not p-nilpotent, G = LP for any
minimal normal subgroup L of G.
For any minimal normal subgroup L of G, if L is p-nilpotent, by the fact that
Lp′charL�G, we have Lp′ ≤ Op′(G) = 1. Thus L is a p-group. Then L ≤ Op(G) = 1
by Step 8, a contradiction. If LP is proper in G, by induction, LP is p-nilpotent, and
so L is p-nilpotent, a contradiction. Thus G = LP for any minimal normal subgroup L
of G.
Step 10. G is a non-abelian simple group.
Take a minimal normal subgroup L of G. If L < G, by Step 9, G = LP. Then G
has a subgroup of index p, contrary to Step 3. Thus G = L is simple.
Step 11. The final contradiction.
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1562 YANGMING LI, SHOUHONG QIAO
Suppose that H is a subgroup of P with |H| = |D| and Q is a Sylow q-subgroup
of G with q 6= p. If H is s-semipermutable in G, then HQg = QgH for any g ∈ G by
the hypotheses and Step 5. Since G is simple by Step 10, G = HQ by Lemma 2.7, a
contradiction. Hence H is s-permutably embedded in G. So H is a Sylow subgroup of
some subnormal subgroup of G. But the subnormal subgroups of G are exactly G and
1, whereas H is a Sylow p-subgroup of neither of them, the final contradiction.
This completes the proof.
Corollary 3.1. Suppose that G is a group. If every non-cyclic Sylow subgroup of
G satisfies ♦1 in G, then G has a Sylow tower of supersolvable type.
Theorem 3.3. Let F be a saturated formation containing U and G a group with a
normal subgroup E such that G/E ∈ F . Suppose that every non-cyclic Sylow subgroup
of E satisfies ♦1 in G. Then G ∈ F .
Proof. Set p ∈ π(E). Suppose that P is a Sylow p-subgroup of E. Since P satisfies
♦1 in G by hypotheses, P satisfies ♦1 in E by Lemma 2.5(a). Applying Corollary 3.1,
we have E has a Sylow tower of supersolvable type. Let q be the largest prime divisor
of |E| and Q ∈ Sylq(E). Then Q � G. Since (G/Q,E/Q) satisfies the hypotheses of
the theorem, by induction, G/Q ∈ F . For any subgroup H of Q with |H| = |D|, since
Q ≤ Oq(G), H is weakly s-permutable in G by Lemma 2.5(e). Hence Q satisfies (∗)′
in G. Since F ∗(Q) = Q by Lemma 2.9, we get G ∈ F by applying Theorem 1.2.
Theorem 3.4. Let F be a saturated formation containing U and G a group with a
normal subgroup E such that G/E ∈ F . Suppose that every non-cyclic Sylow subgroup
of F ∗(E) satisfies ♦1 in G. Then G ∈ F .
Proof. Assume that this theorem is false and let (G,E) be a counterexample with
|G||E| minimal. By Lemma 2.5(a) the hypothesis is still true for (F ∗(E), F ∗(E)), and
so F ∗(E) is supersolvable by Theorem 3.3. Hence F ∗(E) = F (E), by Lemma 2.9(c).
Thus every non-cyclic Sylow subgroup of F ∗(E) satisfies (∗)′ in G. Hence G ∈ F , by
Theorem 1.2.
This completes the proof of the theorem.
4. Some applications. From the definition of weakly s-normal subgroup, we can
see that [4] (Corollaries 5.1 – 5.24) and [6] (Corollaries 4.1 – 4.14) are corollaries of our
Theorems 3.3 and 3.4. Furthermore, we have the following corollaries.
Corollary 4.1. Let F be a saturated formation containing U and let G be a group.
Then G ∈ F if and only if there exists a normal subgroup E such that G/E ∈ F and
all maximal subgroups of any Sylow subgroup of E are either s-permutably embedded
or s-semipermutable or c-normal in G.
Corollary 4.2. Let F be a saturated formation containing U and let G be a group.
Then G ∈ F if and only if there exists a normal subgroup E such that G/E ∈ F and
all maximal subgroups of any Sylow subgroup of F ∗(E) are either s-semipermutable or
c-normal in G.
Corollary 4.3. Let F be a saturated formation containing U and let G be a group.
Then G ∈ F if and only if there exists a normal subgroup E such that G/E ∈ F and
all maximal subgroups of any Sylow subgroup of E are either s-permutably embedded
or c-normal in G.
Corollary 4.4. Let F be a saturated formation containing U and let G be a group.
Then G ∈ F if and only if there exists a normal subgroup E such that G/E ∈ F and
all maximal subgroups of any Sylow subgroup of E are either s-permutably embedded
or s-semipermutable in G.
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ON WEAKLY s-NORMAL SUBGROUPS OF FINITE GROUPS 1563
Corollary 4.5 ([19], Theorem 1). Let F be a saturated formation containing U and
let G be a group. Then G ∈ F if and only if there exists a normal subgroup E such that
G/E ∈ F and all maximal subgroups of any Sylow subgroup of E are s-semipermutable
in G.
Corollary 4.6. Let F be a saturated formation containing U and let G be a group.
Then G ∈ F if and only if there exists a normal subgroup E such that G/E ∈ F and the
cyclic subgroups of prime order or order 4 of F ∗(E) are either s-permutably embedded
or s-semipermutable or c-normal in G.
Corollary 4.7. Let F be a saturated formation containing U and let G be a group.
Then G ∈ F if and only if there exists a normal subgroup E such that G/E ∈ F and all
maximal subgroups of any Sylow subgroup of F ∗(E) are either s-permutably embedded
or s-semipermutable in G.
Corollary 4.8. Let F be a saturated formation containing U and let G be a group.
Then G ∈ F if and only if there exists a normal subgroup E such that G/E ∈ F and the
cyclic subgroups of prime order or order 4 of F ∗(E) are either s-permutably embedded
or c-normal in G.
Corollary 4.9. Let F be a saturated formation containing U and let G be a group.
Then G ∈ F if and only if there exists a normal subgroup E such that G/E ∈ F and
the cyclic subgroups of prime order or order 4 of F ∗(E) are either s-semipermutable
or c-normal in G.
Corollary 4.10 ([19], Theorem 1). Let F be a saturated formation containing U
and let G be a group. Then G ∈ F if and only if there exists a solvable normal
subgroup E such that G/E ∈ F and all maximal subgroups of any Sylow subgroup of
F (E) are s-semipermutable in G.
Corollary 4.11 ([19], Theorem 3). Let F be a saturated formation containing U
and let G be a group. Then G ∈ F if and only if there exists a solvable normal
subgroup E such that G/E ∈ F and the cyclic subgroups of prime order or order 4 of
F (E) are s-semipermutable in G.
Corollary 4.12. Let F be a saturated formation containing U and let G be a
group. Then G ∈ F if and only if there exists a solvable normal subgroup E such that
G/E ∈ F and the cyclic subgroups of F (E) of prime order are either s-permutably
embedded or s-semipermutable in G and the Sylow 2-subgroups of F (E) are abelian.
Corollary 4.13 ([19], Theorem 6). Let F be a saturated formation containing U
and let G be a group. Then G ∈ F if and only if there exists a solvable normal
subgroup E such that G/E ∈ F and the cyclic subgroups of F (E) of prime order are
s-semipermutable in G and the Sylow 2-subgroups of F (E) are abelian.
Theorem 3.2 is also interesting. Using a similar way, we can generalize it as
follows.
Theorem 4.1. Let G be a group, H a normal subgroup of G such that G/H is
p-nilpotent and P a Sylow p-subgroup of H, where p is a prime divisor of |G| with
(|G|, p− 1) = 1. If P satisfies ♦1 in G, then G is p-nilpotent.
Corollary 4.14. Let G be a group, H a normal subgroup of G such that G/H is
p-nilpotent and P a Sylow p-subgroup of H, where p is a prime divisor of |G| with
(|G|, p − 1) = 1. If every maximal subgroup of P is either s-permutably embedded or
s-semipermutable or c-normal in G, then G is p-nilpotent.
Corollary 4.15. Let G be a group, H a normal subgroup of G such that G/H is
p-nilpotent and P a Sylow p-subgroup of H, where p is a prime divisor of |G| with
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1564 YANGMING LI, SHOUHONG QIAO
(|G|, p − 1) = 1. If every maximal subgroup of P is either s-permutably embedded or
s-semipermutable in G, then G is p-nilpotent.
Corollary 4.16. Let G be a group, H a normal subgroup of G such that G/H is
p-nilpotent and P a Sylow p-subgroup of H, where p is a prime divisor of |G| with
(|G|, p− 1) = 1. If P satisfies (∗)′ in G, then G is p-nilpotent.
Corollary 4.17 ([17], Theorem 3.3). Let G be a group, H a normal subgroup of G
such that G/H is p-nilpotent and P a Sylow p-subgroup of H, where p is the minimal
prime dividing the order of G. If every maximal subgroup of P is s-semipermutable in
G, then G is p-nilpotent.
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Received 26.08.09,
after revision — 03.11.11
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