On finite groups with Hall normally embedded Schmidt subgroups
A subgroup H of a finite group G is said to be Hall normally embedded in G if there is a normal subgroup N of G such that H is a Hall subgroup of N. A Schmidt group is a non-nilpotent finite group whose all proper subgroups are nilpotent. In this paper, we prove that if each Schmidt subgroup of a fi...
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Інститут прикладної математики і механіки НАН України
2018
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Kniahina, V.N. Monakhov, V.S. 2023-02-26T12:26:48Z 2023-02-26T12:26:48Z 2018 On finite groups with Hall normally embedded Schmidt subgroups / V.N. Kniahina, V.S. Monakhov // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 1. — С. 90–96. — Бібліогр.: 15 назв. — англ. 1726-3255 2010 MSC: 20E28, 20E32, 20E34. https://nasplib.isofts.kiev.ua/handle/123456789/188376 A subgroup H of a finite group G is said to be Hall normally embedded in G if there is a normal subgroup N of G such that H is a Hall subgroup of N. A Schmidt group is a non-nilpotent finite group whose all proper subgroups are nilpotent. In this paper, we prove that if each Schmidt subgroup of a finite group G is Hall normally embedded in G, then the derived subgroup of G is nilpotent. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On finite groups with Hall normally embedded Schmidt subgroups Article published earlier |
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On finite groups with Hall normally embedded Schmidt subgroups |
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On finite groups with Hall normally embedded Schmidt subgroups Kniahina, V.N. Monakhov, V.S. |
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On finite groups with Hall normally embedded Schmidt subgroups |
| title_full |
On finite groups with Hall normally embedded Schmidt subgroups |
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On finite groups with Hall normally embedded Schmidt subgroups |
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On finite groups with Hall normally embedded Schmidt subgroups |
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on finite groups with hall normally embedded schmidt subgroups |
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Kniahina, V.N. Monakhov, V.S. |
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Kniahina, V.N. Monakhov, V.S. |
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2018 |
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Algebra and Discrete Mathematics |
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Інститут прикладної математики і механіки НАН України |
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A subgroup H of a finite group G is said to be Hall normally embedded in G if there is a normal subgroup N of G such that H is a Hall subgroup of N. A Schmidt group is a non-nilpotent finite group whose all proper subgroups are nilpotent. In this paper, we prove that if each Schmidt subgroup of a finite group G is Hall normally embedded in G, then the derived subgroup of G is nilpotent.
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1726-3255 |
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https://nasplib.isofts.kiev.ua/handle/123456789/188376 |
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On finite groups with Hall normally embedded Schmidt subgroups / V.N. Kniahina, V.S. Monakhov // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 1. — С. 90–96. — Бібліогр.: 15 назв. — англ. |
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“adm-n3” — 2018/10/20 — 9:02 — page 90 — #96
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 26 (2018). Number 1, pp. 90–96
c© Journal “Algebra and Discrete Mathematics”
On finite groups with Hall normally embedded
Schmidt subgroups
Viktoryia N. Kniahina and Victor S. Monakhov
Communicated by L. A. Kurdachenko
To the 70th anniversary of Academician
of the National Academy of Sciences of Belarus V. I. Yanchevskii
Abstract. A subgroup H of a finite group G is said to
be Hall normally embedded in G if there is a normal subgroup N
of G such that H is a Hall subgroup of N . A Schmidt group is a
non-nilpotent finite group whose all proper subgroups are nilpotent.
In this paper, we prove that if each Schmidt subgroup of a finite
group G is Hall normally embedded in G, then the derived subgroup
of G is nilpotent.
1. Introduction
All groups in this paper are finite. We use the standard notation and
terminology of [1, 2].
A Schmidt group is a non-nilpotent group in which every proper
subgroup is nilpotent. O.Y. Schmidt [3] initiated the investigations of
such groups. He proved that a Schmidt group is biprimary (i. e. its order is
divided by only two different primes), one of its Sylow subgroups is normal
and other one is cyclic. In [3], it was also specified the index system of the
chief series of a Schmidt group. Reviews on the structure of the Schmidt
2010 MSC: 20E28, 20E32, 20E34.
Key words and phrases: finite group, Hall subgroup, normal subgroup, derived
subgroup, nilpotent subgroup.
“adm-n3” — 2018/10/20 — 9:02 — page 91 — #97
V. N. Kniahina, V. S. Monakhov 91
groups and their applications in the theory of finite groups are available
in [4, 5].
Since every non-nilpotent group contains a Schmidt subgroup, Schmidt
groups are universal subgroups of groups. So naturally the properties of
Schmidt subgroups contained in a group have a significant influence on
the group structure. Groups with some restrictions on Schmidt subgroups
was investigated in many papers. For example, groups with subnormal
Schmidt subgroups were studied in [6]–[8], and groups with Hall Schmidt
subgroups were described in [9].
The normal closure of a subgroup H in a group G is the smallest
normal subgroup of G containing H. It is clear that the normal closure
HG = 〈Hx | x ∈ G〉 =
⋂
H6N⊳G
N
coincides with the intersection of all normal subgroups of G containing H .
A subgroup H of a group G is said to be Hall normally embedded in
G if there is a normal subgroup N of G such that H 6 N and H is a Hall
subgroup of N , i.e., (|H|, |N : H|) = 1. In this situation the subgroup H
is a Hall subgroup of HG. It is clear that all normal and all Hall subgroups
of G are Hall normally embedded in G.
Groups in which some subgroups are normally embedded were studied,
for example, in [10]–[13].
In this paper, we study groups with Hall normally embedded Schmidt
subgroups. The following theorem is proved.
Theorem. If each Schmidt subgroup of a group G is Hall normally em-
bedded in G, then the derived subgroup of G is nilpotent.
2. Preliminaries
Throughout this paper, p and q are always different primes. Recall
that a p-closed group is a group with a normal Sylow p-subgroup, and a
p-nilpotent group is a group of order pam, where p does not divide m, with
a normal subgroup of order m. A pd-group is a group of the order divided
by p. A group of order paqb, where a and b are non-negative integers, is
called a {p, q}-group.
If q divides pn− 1 and does not divide pn1 − 1 for all 1 6 n1 < n, then
we say that the positive integer n is the order of p modulo q.
Let G be a group. We denote by π(G) the set of all prime divisors
of the order of G. We use Z(G), Φ(G) and F (G) to denote the center,
“adm-n3” — 2018/10/20 — 9:02 — page 92 — #98
92 On finite groups with Hall normally embedded. . .
the Frattini subgroup and the Fitting subgroup of G, respectively. As
usual, Op(X) and Op′(X) are the largest normal p- and p′-subgroups of X,
respectively. We denote by [A]B a semidirect product of two subgroups A
and B with a normal subgroup A. The symbol � indicates the end of the
proof.
We need the following properties of Schmidt groups.
Lemma 1 ([3,5]). Let S be a Schmidt group. Then the following statements
hold:
(1) π(S) = {p, q}, S = [P ]〈y〉, where P is a normal Sylow p-subgroup,
〈y〉 is a non-normal Sylow q-subgroup, yq ∈ Z(S);
(2) P/Φ(P ) is a minimal normal subgroup of G/Φ(P ), Φ(P ) = P ′ 6
Z(G);
(3) |P/Φ(P )| = pn, n is the order of p modulo q.
Following [6], a Schmidt group with a normal Sylow p-subgroup and a
non-normal cyclic Sylow q-subgroup is called an S〈p,q〉-group. So if G is
an S〈p,q〉-group, then G = [P ]Q, where P is a normal Sylow p-subgroup
and Q is a non-normal cyclic Sylow q-subgroup.
Lemma 2 ([6, Lemma 6]). (1) If a group G has no p-closed Schmidt
subgroups, then G is p-nilpotent.
(2) If a group G has no 2-nilpotent Schmidt 2d-subgroups, then G is
2-closed.
(3) If a p-soluble group G has no p-nilpotent Schmidt pd-subgroups, then
G is p-closed.
Lemma 3. Let A be a subgroup of a group G such that A is a Hall
subgroup of AG.
(1) If H is a subgroup of G, A 6 H, then A is a Hall subgroup of AH .
(2) If N is a normal subgroup of G, then AN/N is a Hall subgroup of
(AN/N)(G/N).
Proof. 1. By the hypothesis, A is a Hall subgroup of AG and A 6 H ∩AG.
Since AG is normal in G, it follows that H ∩ AG is normal in H. So
AH 6 H ∩AG 6 AG and A is a Hall subgroup of AH .
2. Since AGN is normal in G and AN 6 AGN , so (AN/N)(G/N) 6
AGN/N . By the hypothesis, A is a Hall subgroup of AG, thus AN/N
is a Hall subgroup of AGN/N . Therefore, AN/N is a Hall subgroup of
(AN/N)(G/N).
“adm-n3” — 2018/10/20 — 9:02 — page 93 — #99
V. N. Kniahina, V. S. Monakhov 93
Lemma 4. Let K and D be subgroups of a group G such that D is normal
in K. If K/D is an S〈p,q〉-subgroup, then each minimal supplement L to
D in K has the following properties:
(1) L is a p-closed {p, q}-subgroup;
(2) all proper normal subgroups of L are nilpotent;
(3) L includes an S〈p,q〉-subgroup [P ]Q such that D does not include Q
and L = ([P ]Q)L = QL;
(4) if [P ]Q is a Hall subgroup of ([P ]Q)G, then L = [P ]Q.
Proof. Assertions (1)–(3) were established in [6, Lemma 2]. Let us verify
assertion (4). If [P ]Q is a Hall subgroup of ([P ]Q)G, then [P ]Q is a Hall
subgroup of ([P ]Q)L = L by Lemma 3 (1), and L = [P ]Q.
Lemma 5. If H is a subgroup of a group G generated by all S〈p,q〉-sub-
groups of G, then G/H has no S〈p,q〉-subgroups.
Proof. Assume the contrary. Suppose that A/H is a S〈p,q〉-subgroup of
G/H . By Lemma 4, in A there is an S〈p,q〉-subgroup S such that SAH = A.
However, SA 6 H by the choice of H, i. e A = H, a contradiction.
Lemma 6. Let each S〈p,q〉-subgroup of a group G be Hall normally em-
bedded in G.
(1) If H is a subgroup of G, then each S〈p,q〉-subgroup of H is Hall
normally embedded in H.
(2) If N is a normal subgroup of G, then each S〈p,q〉-subgroup of G/N
is Hall normally embedded in G/N .
Proof. 1. Let A be an S〈p,q〉-subgroup of H . Therefore, A is an S〈p,q〉-sub-
group of G. By the hypothesis,A is a Hall subgroup of AG. By Lemma 3 (1),
A is a Hall subgroup of AH .
2. Let K/N be an S〈p,q〉-subgroup of G/N , and let L be a minimal
supplement to N in K. By Lemma 4 (4), L is an S〈p,q〉-subgroup, therefore,
L is Hall normally embedded in G. By Lemma 3 (2), LN/N = K/N is
Hall normally embedded in G/N .
Lemma 7. Let G be a p-soluble group and lp(G) > 1. If lp(H) 6 1 and
lp(G/K) 6 1 for each H < G, 1 6= K ⊳G, then the following hold:
(1) Φ(G) = Op′(G) = 1;
(2) G has a unique minimal normal subgroup N=F (G)=Op(G)=CG(N);
(3) lp(G) = 2;
(4) G = [N ]S, where S=[Q]P is a p-nilpotent Schmidt subgroup, |P |=p.
“adm-n3” — 2018/10/20 — 9:02 — page 94 — #100
94 On finite groups with Hall normally embedded. . .
Proof. Assertions (1)–(2) follow from [2, VI.6.9]. As lp(N) = 1 and
lp(G/N) 6 1 we have lp(G) = 2. It remains to prove assertion (4). Since
G is a p-soluble non-p-closed group, we conclude from Lemma 2 (3) that
in G there is an S〈q,p〉-subgroup S = [Q]P for some q ∈ π(G). Suppose
that NS is a proper subgroup of G. Then Op′(NS) 6 CG(N) = N . Thus,
Op′(NS) = 1. By the hypothesis, lp(NS) = 1, so NS is p-closed. This
contradicts the fact that S is not p-closed. Therefore, NS = G. Moreover
N ∩S⊳G, N ∩S = 1, and S is a maximal subgroup of G. Since Op(S) = 1,
it follows from Lemma 1 that |P | = p.
Lemma 8. If each p-nilpotent Schmidt pd-subgroup of a p-soluble group G
is Hall normally embedded in G, then lp(G) 6 1.
Proof. Let G be a counterexample of minimal order. By Lemma 6, each
proper subgroup and each non-trivial quotient group of G have a p-length 6
1. By Lemma 7,
G = [N ]S, Φ(G) = Op′(G) = 1, N = Op(G) = F (G) = CG(N),
where S = [Q]P is a maximal subgroup of G and is an S〈p,q〉-subgroup
for some q ∈ π(G). By the hypothesis, S is a Hall subgroup of SG. Since
SG = G, it follows that N is a p′-subgroup, a contradiction.
Lemma 9. Let n > 2 be a positive integer, let r be a prime, and let π be
the set of primes t such that t divides rn − 1 but t does not divide rn1 − 1
for all 1 6 n1 < n. Then the group GL(n, r) contains a cyclic π-Hall
subgroup.
Proof. The group G = GL(n, r) is of order
rn(n−1)/2(rn − 1)(rn−1 − 1) . . . (r2 − 1)(r − 1).
By Theorem II.7.3 [2], G contains a cyclic subgroup T of order rn − 1. Its
π-Hall subgroup Tπ is a π-Hall subgroup of G, because t does not divide
rn1 − 1 for all t ∈ π and all 1 6 n1 < n.
3. Proof of the theorem
We proceed by induction on the order of G. First, we verify that G
is soluble. Assume the contrary. It follows that G is not 2-closed, and by
Lemma 2 (2), in G there exists a 2-nilpotent Schmidt subgroup S = [P ]Q
of even order, where P is a Sylow p-subgroup of order p > 2, Q is a
“adm-n3” — 2018/10/20 — 9:02 — page 95 — #101
V. N. Kniahina, V. S. Monakhov 95
cyclic Sylow 2-subgroup. By the hypothesis, S is a Hall subgroup of SG,
therefore, Q is a Sylow 2-subgroup of SG, and SG is 2-nilpotent. Thus,
S 6 SG 6 R(G). Here R(G) is the largest normal soluble subgroup of G.
Since S is arbitrary, we conclude that all 2-nilpotent Schmidt subgroups of
even order are contained in R(G). By Lemma 5, the quotient group G/R(G)
has no 2-nilpotent Schmidt subgroups of even order. By Lemma 2 (2), the
quotient group G/R(G) is 2-closed, therefore, G is soluble.
Note that the derived subgroup G′ is nilpotent if and only if G ∈ NA.
Here N, A and E are the formations of all nilpotent, abelian and finite
groups, respectively, and
NA = { G ∈ E | G′ ∈ N}
is the formation product of N and A. According to [14, p. 337], NA is
an s-closed saturated formation. The quotient group G/N ∈ NA for each
non-trivial normal subgroup N of G by Lemma 6 (2). A simple check
shows that
G = [N ]M, N = Op(G) = F (G) = CG(N), |N | = pn, MG = 1,
where N is a unique minimal normal subgroup of G, M is a maximal
subgroup of G. In view of Lemma 7, N is a Sylow p-subgroup of G.
Let π = π(M) = π(G) \ {p}, r ∈ π, and let R be a Sylow r-subgroup
of G. Since N = CG(N), we obtain from Lemma 2 (1) that in [N ]R
there is an S〈p,r〉-subgroup [P1]R1. By the hypothesis, [P1]R1 is a Hall
subgroup of ([P1]R1)
G, therefore, P1 is a Sylow p-subgroup of ([P1]R1)
G.
Since N 6 ([P1]R1)
G and N is a Sylow p-subgroup of G, it follows that
N = P1. By Lemma 1, n is the order of p modulo r. But r is an arbitrary
number from π, so n is the order of p modulo q for all q ∈ π. The group
M ≃ G/N is isomorphic to a subgroup of GL(n, p), which contains a
cyclic Hall π-subgroup H by Lemma 9. In view of Theorem 5.3.2 [15], M
is contained in a subgroup Hx, x ∈ GL(n, p). Therefore, M is cyclic.
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Contact information
V. N. Kniahina,
V. S. Monakhov
Department of Mathematics, Francisk Skorina
Gomel State University, Sovetskaya str., 104,
Gomel 246019, Belarus
E-Mail(s): Knyagina@inbox.ru,
Victor.Monakhov@gmail.com
Received by the editors: 20.04.2018.
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