On p-nilpotency of finite group with normally embedded maximal subgroups of some Sylow subgroups
Let G be a finite group and P be a p-subgroup of G. If P is a Sylow subgroup of some normal subgroup of G, then we say that P is normally embedded in G. Groups with normally embedded maximal subgroups of Sylow p-subgroup, where (|G|, p − 1) = 1, are studied. In particular, the p-nilpotency of such g...
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| Цитувати: | On p-nilpotency of finite group with normally embedded maximal subgroups of some Sylow subgroups / A. Trofimuk // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 1. — С. 139–146. — Бібліогр.: 9 назв. — англ. |
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Trofimuk, A. 2023-03-03T16:13:45Z 2023-03-03T16:13:45Z 2020 On p-nilpotency of finite group with normally embedded maximal subgroups of some Sylow subgroups / A. Trofimuk // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 1. — С. 139–146. — Бібліогр.: 9 назв. — англ. 1726-3255 DOI:10.12958/adm1128 2010 MSC: 20D10 https://nasplib.isofts.kiev.ua/handle/123456789/188509 Let G be a finite group and P be a p-subgroup of G. If P is a Sylow subgroup of some normal subgroup of G, then we say that P is normally embedded in G. Groups with normally embedded maximal subgroups of Sylow p-subgroup, where (|G|, p − 1) = 1, are studied. In particular, the p-nilpotency of such groups is proved. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On p-nilpotency of finite group with normally embedded maximal subgroups of some Sylow subgroups Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
On p-nilpotency of finite group with normally embedded maximal subgroups of some Sylow subgroups |
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On p-nilpotency of finite group with normally embedded maximal subgroups of some Sylow subgroups Trofimuk, A. |
| title_short |
On p-nilpotency of finite group with normally embedded maximal subgroups of some Sylow subgroups |
| title_full |
On p-nilpotency of finite group with normally embedded maximal subgroups of some Sylow subgroups |
| title_fullStr |
On p-nilpotency of finite group with normally embedded maximal subgroups of some Sylow subgroups |
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On p-nilpotency of finite group with normally embedded maximal subgroups of some Sylow subgroups |
| title_sort |
on p-nilpotency of finite group with normally embedded maximal subgroups of some sylow subgroups |
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Trofimuk, A. |
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Trofimuk, A. |
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2020 |
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Algebra and Discrete Mathematics |
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Інститут прикладної математики і механіки НАН України |
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Article |
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Let G be a finite group and P be a p-subgroup of G. If P is a Sylow subgroup of some normal subgroup of G, then we say that P is normally embedded in G. Groups with normally embedded maximal subgroups of Sylow p-subgroup, where (|G|, p − 1) = 1, are studied. In particular, the p-nilpotency of such groups is proved.
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1726-3255 |
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https://nasplib.isofts.kiev.ua/handle/123456789/188509 |
| citation_txt |
On p-nilpotency of finite group with normally embedded maximal subgroups of some Sylow subgroups / A. Trofimuk // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 1. — С. 139–146. — Бібліогр.: 9 назв. — англ. |
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| fulltext |
“adm-n1” — 2020/5/14 — 19:35 — page 139 — #147
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 29 (2020). Number 1, pp. 139–146
DOI:10.12958/adm1128
On p-nilpotency of finite group with normally
embedded maximal subgroups of some Sylow
subgroups
A. Trofimuk
Communicated by A. Yu. Olshanskii
Abstract. Let G be a finite group and P be a p-subgroup
of G. If P is a Sylow subgroup of some normal subgroup of G,
then we say that P is normally embedded in G. Groups with nor-
mally embedded maximal subgroups of Sylow p-subgroup, where
(|G|, p− 1) = 1, are studied. In particular, the p-nilpotency of such
groups is proved.
Introduction
All groups considered in this paper will be finite. Our notation is
standard and taken mainly from [1], [2].
Let M(G) be the set of all maximal subgroups of Sylow subgroups of
a group G. One of the first results related to the study of the structure of
a group with given restrictions on M(G) belongs to Srinivasan, see [3].
In particular, in [3] it is proved that a group G is supersolvable, if every
subgroup of M(G) is normal in G. Subsequently, groups with restrictions
on subgroups of M(G) have been studied in the works of many authors,
see the literature in [4].
A subgroup H of G is said to be S-embedded in G, see [5], if G
has a normal subgroup N such that HN is S-permutable in G and
2010 MSC: 20D10.
Key words and phrases: p-supersolvable group, normally embedded subgroup,
maximal subgroup, Sylow subgroup.
https://doi.org/10.12958/adm1128
“adm-n1” — 2020/5/14 — 19:35 — page 140 — #148
140 On p-nilpotency of finite group
H ∩ N 6 HsG, where HsG is the largest S-permutable subgroup of G
contained in H. In the paper [5] the structure of the groups depending
on S-embedded subgroups is studied. In particular, the p-nilpotency of
a group G for which every subgroup of M(P ) is S-embedded in G, where
P is a Sylow p-subgroup of G and p ∈ π(G) such that (|G|, p − 1) = 1
follows from [5, Theorem 2.3].
In the present paper, we study another generalization of normality.
Definition. A subgroup H of a group G is said normally embedded in
G, if for every Sylow subgroup P of H, there is a normal subgroup K of
G such that P is Sylow subgroup of K, see [6, I.7.1].
A series of results related to the structure of a group with normally
embedded subgroups is presented in [6].
The following examples show that S-embedded and normally embedded
are different concepts.
In the symmetric group S5 of degree 5 some maximal subgroup H
of a Sylow 2-subgroup is a Sylow 2-subgroup in the normal alternating
subgroup A5 of degree 5, i.e. H is normally embedded in S5. But, H is
not S-embedded. In the alternating group A4 of degree 4 some maximal
subgroup M of a Sylow 2-subgroup is not normally embedded in A4. But,
M is S-embedded.
In this paper, the structure of a group G under the condition that
every subgroup of M(P ) is normally embedded in G is studied, where P
is a Sylow p-subgroup of G and p ∈ π(G) such that (|G|, p− 1) = 1.
The following theorem is proved.
Theorem. Let G be a group, H be a normal subgroup of G such that G/H
is p-nilpotent and P be a Sylow p-subgroup of H, where p ∈ π(G) with
(|G|, p− 1) = 1. If every subgroup of M(P ) is normally embedded in G,
then G is p-nilpotent.
1. Preliminaries
In this section we collect lemmas used in the proof of the main theorem
presented in Section 2.
The Fitting subgroup and the Frattini subgroup of G are denoted by
F (G) and Φ(G), respectively; we write Zm for a cyclic group of orders m;
Op(G) and Op′(G) denote the greatest normal p-subgroup of G and the
greatest normal p′-subgroup of G, respectively. By π(G) denote the set
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A. Trofimuk 141
of all prime divisors of the order of G; by HG denote the normal closure
of a subgroup H in a group G, i.e. the smallest normal subgroup of G
containing H . We write H neG for normally embedded subgroup H of G
and G = [A]B for the semidirect product of some subgroups A and B
with the normal subgroup A.
If the orders of chief factors of G are either equal to p or not divisible
on p then G is called p-supersolvable. We denote by pU the class of all
p-supersolvable groups. A group that has a normal Sylow p-subgroup is
called p-closed and a group that has a normal p′-Hall subgroup is called
p-nilpotent.
Let G be a group of order pa1
1
pa2
2
. . . pakk , where p1 > p2 > . . . > pk.
We say that G has an ordered Sylow tower of supersolvable type if there
exists a series
1 = G0 < G1 < G2 < . . . < Gk−1 < Gk = G
of normal subgroups of G such that Gi/Gi−1 is isomorphic to a Sylow
pi-subgroup of G for each i = 1, 2, . . . , k.
Lemma 1 ([6, I.7.3]). Let U be a normally embedded p-subgroup of
a group G, K a normal subgroup of G. Then:
(1) if U 6 H 6 G, then U neH;
(2) UK/K neG/K;
(3) U ∩K neG;
(4) if K is a p-group, then UK neG and U ∩K is normal in G;
(5) Ug neH for all g ∈ G.
Lemma 2. Let H be a normal subgroup of G and every maximal subgroup
of Sylow p-subgroup of H is normally embedded in G. If N is normal
in G, then every maximal subgroup of every Sylow p-subgroup of HN/N
is normally embedded in G/N . In particular, if N is normal in G and
every maximal subgroup of Sylow p-subgroup of G is normally embedded
in G, then every maximal subgroup of every Sylow p-subgroup of G/N is
normally embedded in G/N .
Proof. By Lemma 1 (5), it follows that X1 is normally embedded in G
for any Sylow p-subgroup X of H and any maximal subgroup X1 of X.
Let P1 = X/N is a maximal subgroup of Sylow p-subgroup P of HN/N .
Then N 6 X 6 HN and there exists a Sylow p-subgroup P in HN such
that P = PN/N . By [1, VI.4.6], there exist the Sylow p-subgroups Hp
in H and Np in N such that P = HpNp, hence P = HpN/N . Further,
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142 On p-nilpotency of finite group
N 6 X < PN 6 HpN and X = (X ∩Hp)N by Dedekind’s identity. Since
Hp ∩N = X ∩Hp ∩N , we have
p = |P : P1| = |HpN/N : X/N | = |HpN : X|
= |HpN : (X ∩Hp)N | =
|Hp||N ||X ∩Hp ∩N |
|Hp ∩N ||X ∩Hp||N |
= |Hp : X ∩Hp|.
So, X ∩ Hp is a maximal subgroup in Hp. By hypothesis, X ∩ Hp is
normally embedded in G. By Lemma 1 (2), (X ∩ Hp)N/N = X/N is
normally embedded in G/N .
For H = G we obtain the second part of the lemma.
Lemma 3 ([7, Lemma 5]). Let G be a p-solvable group. Assume that G
does not belong to pU, but G/K ∈ pU for all non-trivial normal subgroups
K of G. Then:
(1) Z(G) = Op′(G) = Φ(G) = 1;
(2) G contains a unique minimal normal subgroup N , N = F (G) =
Op(G) = CG(N);
(3) G is primitive; G = [N ]M , where M is maximal in G with trivial
core;
(4) N is an elementary Abelian subgroup of order pn, n > 1;
(5) if M is Abelian, then M is cyclic of order dividing pn − 1, and n is
the smallest natural number such that pn ≡ 1 (mod |M |).
A non-nilpotent group whose proper subgroups are all nilpotent is
called a Schmidt group.
Lemma 4 ([8]). Let S be a Schmidt group. Then:
(1) S = [P ]Q, where P is a normal Sylow p-subgroup, Q is a non-normal
Sylow q-subgroup, p and q are distinct primes;
(2) Q = 〈y〉 is cyclic and yq ∈ Z(S);
(3) |P/P ′| = pm, where m is the order of p modulo q;
(4) the chief series of S has the following system of indexes: p, p, . . . , p,
pm, q, . . . , q; number of indexes equal to p coincides with n, where
pn = |P ′|; number of indexes equal to q coincides with b, where
qb = |Q|.
Lemma 5. Let p ∈ π(G) and (|G|, p− 1) = 1. Then G is p-supersolvable
if and only if G is p-nilpotent. In particular, if a Sylow p-subgroup is cyclic,
then G is p-nilpotent.
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A. Trofimuk 143
Proof. It is clear that every p-nilpotent group is p-supersolvable. Con-
versely. Let G be a group of the smallest order such that G is p-super-
solvable, but is not p-nilpotent. Let H be an arbitrary proper subgroup
of G. Then H is p-supersolvable and (|H|, p− 1) = 1. Therefore in view
of the choice G, the subgroup H is p-nilpotent and G is a minimal non-
p-nilpotent group. By [9, Theorem 10.3.3], G is a Schmidt group. By
Lemma 4 (1), G = [P ]Q, where P is a Sylow p-subgroup and Q is a cyclic
Sylow q-subgroup. Since G is p-supersolvable, then by Lemma 4 (4), the
order of p modulo q is equal 1, i.e. m = 1. Hence q divides p− 1. This is
a contradiction.
In particular, if a Sylow p-subgroup is cyclic, then G is p-supersolvable.
Then G is p-nilpotent by what has been proved above. The lemma is
proved.
Corollary 1. Let p be the smallest prime of π(G). Then G is p-super-
solvable if and only if G is p-nilpotent.
Example 1. The symmetric group G = S3 of degree 3 is 3-supersolvable,
but is not 3-nilpotent. Hence, the condition (|G|, p− 1) = 1 in Lemma 5
can not be removed.
Example 2. A group G = Z5 × ([Z7]Z3) is 5-supersolvable and is 5-
nilpotent. In addition, (|G|, 5− 1) = 1, and the prime divisor 5 of |G| is
not the smallest.
Evidently, if a p-subgroup P of G is normally embedded in G, then P
is a Sylow subgroup of PG.
Lemma 6. Let G be a group, Φ(G) = 1, P be a Sylow subgroup of G with
unprimary order and N be a unique minimal normal subgroup of G. If
every subgroup of M(P ) is normally embedded in G and N is Abelian,
then N is not contained in P .
Proof. Suppose that N 6 P . If N = P , then by hypothesis, every maximal
subgroup S of P is normally embedded in G. Then by Lemma 1 (4), S
is normal in G. Since the order of P is not equal to a prime, we have
a contradiction with the fact that N is a minimal normal subgroup in G.
In the following we assume that N < P . Since Φ(G) = 1, it follows
that there exists a maximal subgroup M of G such that N is not contained
in M . Hence G = NM . By [2, Lemma 2.36], N ∩M = 1 and G = [N ]M .
Then by Dedekind’s identity, P = P ∩ [N ]M = [N ](P ∩ M), where
“adm-n1” — 2020/5/14 — 19:35 — page 144 — #152
144 On p-nilpotency of finite group
P ∩M 6= 1. Let T be a maximal subgroup of P such that P ∩M 6 T .
Since N is a unique minimal normal subgroup of G, it follows that N 6 TG.
Now, P = NT 6 TG, but by hypothesis, T is a Sylow subgroup of TG,
a contradiction.
Lemma 7. Let P be a Sylow p-subgroup of G. If every subgroup of M(P )
is normally embedded in G and (|G|, p− 1) = 1, then G is p-nilpotent.
Proof. We use induction on the order of G. Since (|G/N |, p− 1) = 1 and
by Lemma 2, every maximal subgroup of every Sylow p-subgroup of G/N
is normally embedded in G/N for any normal subgroup N 6= 1 of G, then
all quotients of G satisfy the hypotheses of the lemma.
By the inductive hypothesis, Op′(G) = 1. Since the class of all p-
nilpotent groups is a saturated formation, then Φ(G) = 1 and N =
F (G) = Op(G) is a unique minimal normal subgroup G. Hence there is
a Sylow p-subgroup R of G such that N ⊆ R. Since R and P are conjugate
in G, then by Lemma 1 (5), it follows that every maximal subgroup of R
is normally embedded in G. If |R| = p, then G is p-nilpotent by Lemma 5.
Therefore, we further assume that |R| > p. By Lemma 6,N is not contained
in R. This is a contradiction. The lemma is proved.
2. Proof of the theorem
In view of Lemma 5, we prove that G is p-supersolvable.
By Lemma 1 (1), every maximal subgroup of Sylow p-subgroup P of
H is normally embedded in H and (|H|, p− 1) = 1. By Lemma 7, H is
p-nilpotent. Since by hypothesis, G/H is p-nilpotent, then G is p-solvable.
We use induction on the order of G. Let N be an arbitrary non-trivial
normal subgroup of G. Clearly, HN/N is normal in G/N and
(G/N)/(HN/N) ∼= G/(HN) ∼= (G/H)/(HN/H)
is p-nilpotent. Besides, by Lemma 2, every maximal subgroup of ev-
ery Sylow p-subgroup of HN/N is normally embedded in G/N and
(|G/N |, p− 1) = 1. Hence the quotients G/N satisfy the hypotheses of
the theorem.
By the inductive hypothesis, G/N is p-supersolvable. By Lemma 3,
Z(G) = Op′(G) = Φ(G) = 1, G contains a unique minimal normal
subgroup
N = F (G) = Op(G) = CG(N), G = [N ]M,
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A. Trofimuk 145
N is an elementary Abelian subgroup of order pn, n > 1, M is a maximal
subgroup of G.
Since N 6 H , then N is contained in every Sylow p-subgroup P of H .
By Lemma 6, we have a contradiction. The theorem is proved.
Corollary 2. Let G be a group, H be a normal subgroup of group G such
that G/H is p-nilpotent and P be a Sylow p-subgroup of H, where p is the
smallest in π(G). If every subgroup of M(P ) is normally embedded in G,
then G is p-nilpotent.
Corollary 3. Let G be a group and P be a Sylow p-subgroup of G, where
p ∈ π(G) with (|G|, p − 1) = 1. If every subgroup of M(P ) is normally
embedded in G, then G is p-nilpotent.
Corollary 4. Let G be a group and P be a Sylow p-subgroup of G, where
p is the smallest in π(G). If every subgroup of M(P ) is normally embedded
in G, then G is p-nilpotent.
Corollary 5. Let G be a group. If every subgroup of M(G) is normally
embedded in G, then G possesses an ordered Sylow tower of supersolvable
type.
Proof. Let p be the smallest prime of π(G) and P be a Sylow p-subgroup
of G. Then by hypothesis, every subgroup of M(P ) is normally embed-
ded in G. By Corollary 4, G is p-nilpotent. By Lemma 1 (1) and the
inductive hypothesis, a Hall p′-subgroup of G has an ordered Sylow tower
of supersolvable type. Consequently, G has an ordered Sylow tower of
supersolvable type.
References
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[3] S. Srinivasan, Two sufficient conditions for supersolvability of finite groups, Israel J.
Math., 35, 1980, pp.210–214.
[4] V. S. Monakhov, A. A. Trofimuk, Finite groups with subnormal non-cyclic subgroups,
J. Group Theory, 17(5), 2014, pp.889–895.
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49(4), 2010, pp.433–450.
[6] K. Doerk and T. Hawkes, Finite soluble groups. Berlin-New York: Walter de Gruyter,
1992.
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146 On p-nilpotency of finite group
[7] V. S. Monakhov, I. K. Chirik. On the p-supersolvability of a finite factorizable group
with normal factors, Proceedings of the Institute of Mathematics and Mechanics
(Trudy Instituta Matematiki I Mekhaniki), 21(3), 2015, pp.256–267.
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Contact information
Alexander Trofimuk Department of Mathematics, Gomel Francisk
Skorina State University, Gomel 246019,
Belarus
E-Mail(s): alexander.trofimuk@gmail.com
Received by the editors: 21.04.2018.
A. Trofimuk
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