Finite groups with semi-subnormal Schmidt subgroups
A Schmidt group is a non-nilpotent group in which every proper subgroup is nilpotent. A subgroup A of a group G is semi-normal in G if there exists a subgroup B of G such that G = AB and AB1 is a proper subgroup of G for every proper subgroup B1 of B. If A is either subnormal in G or is semi-normal...
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Kniahina, V.N. Monakhov, V.S. 2023-03-03T15:51:45Z 2023-03-03T15:51:45Z 2020 Finite groups with semi-subnormal Schmidt subgroups / V.N. Kniahina, V.S. Monakhov // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 1. — С. 66–73. — Бібліогр.: 17 назв. — англ. 1726-3255 DOI:10.12958/adm1376 2010 MSC: 20E28, 20E32, 20E34 https://nasplib.isofts.kiev.ua/handle/123456789/188502 A Schmidt group is a non-nilpotent group in which every proper subgroup is nilpotent. A subgroup A of a group G is semi-normal in G if there exists a subgroup B of G such that G = AB and AB1 is a proper subgroup of G for every proper subgroup B1 of B. If A is either subnormal in G or is semi-normal in G, then A is called a semi-subnormal subgroup of G. In this paper, we establish that a group G with semi-subnormal Schmidt {2, 3}-subgroups is 3-soluble. Moreover, if all 5-closed Schmidt {2, 5}-subgroups are semi-subnormal in G, then G is soluble. We prove that a group with semi-subnormal Schmidt subgroups is metanilpotent. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Finite groups with semi-subnormal Schmidt subgroups Article published earlier |
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Finite groups with semi-subnormal Schmidt subgroups |
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Finite groups with semi-subnormal Schmidt subgroups Kniahina, V.N. Monakhov, V.S. |
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Finite groups with semi-subnormal Schmidt subgroups |
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Finite groups with semi-subnormal Schmidt subgroups |
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Finite groups with semi-subnormal Schmidt subgroups |
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Finite groups with semi-subnormal Schmidt subgroups |
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finite groups with semi-subnormal 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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Algebra and Discrete Mathematics |
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A Schmidt group is a non-nilpotent group in which every proper subgroup is nilpotent. A subgroup A of a group G is semi-normal in G if there exists a subgroup B of G such that G = AB and AB1 is a proper subgroup of G for every proper subgroup B1 of B. If A is either subnormal in G or is semi-normal in G, then A is called a semi-subnormal subgroup of G. In this paper, we establish that a group G with semi-subnormal Schmidt {2, 3}-subgroups is 3-soluble. Moreover, if all 5-closed Schmidt {2, 5}-subgroups are semi-subnormal in G, then G is soluble. We prove that a group with semi-subnormal Schmidt subgroups is metanilpotent.
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| issn |
1726-3255 |
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https://nasplib.isofts.kiev.ua/handle/123456789/188502 |
| citation_txt |
Finite groups with semi-subnormal Schmidt subgroups / V.N. Kniahina, V.S. Monakhov // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 1. — С. 66–73. — Бібліогр.: 17 назв. — англ. |
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“adm-n1” — 2020/5/14 — 19:35 — page 66 — #74
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 29 (2020). Number 1, pp. 66–73
DOI:10.12958/adm1376
Finite groups with semi-subnormal Schmidt
subgroups
V. N. Kniahina and V. S. Monakhov
Communicated by I. Ya. Subbotin
For the 70th anniversary of L.A. Kurdachenko
Abstract. A Schmidt group is a non-nilpotent group in
which every proper subgroup is nilpotent. A subgroup A of a group
G is semi-normal in G if there exists a subgroup B of G such that
G = AB and AB1 is a proper subgroup of G for every proper
subgroup B1 of B. If A is either subnormal in G or is semi-normal
in G, then A is called a semi-subnormal subgroup of G. In this
paper, we establish that a group G with semi-subnormal Schmidt
{2, 3}-subgroups is 3-soluble. Moreover, if all 5-closed Schmidt {2, 5}-
subgroups are semi-subnormal in G, then G is soluble. We prove that
a group with semi-subnormal Schmidt subgroups is metanilpotent.
1. Introduction
All groups in this paper are finite. A Schmidt group is a non-nilpotent
group in which every proper subgroup is nilpotent. These groups were
first considered by O. Y. Schmidt [1]. He proved that a Schmidt group is
biprimary, one of its Sylow subgroups is normal and another one is cyclic.
Reviews about the structure of Schmidt groups and their applications in
the theory of finite groups are available in [2], [3].
Since every non-nilpotent group contains a Schmidt subgroup, Schmidt
groups are universal subgroups of finite groups. So the properties of
2010 MSC: 20E28, 20E32, 20E34.
Key words and phrases: finite soluble group, Schmidt subgroup, semi-normal
subgroup, subnormal subgroup.
https://doi.org/10.12958/adm1376
“adm-n1” — 2020/5/14 — 19:35 — page 67 — #75
V. N. Kniahina, V. S. Monakhov 67
Schmidt subgroups contained in a group have a significant influence on
the group structure. Groups with some restrictions on Schmidt subgroups
were investigated in many papers. For example, groups with subnormal
Schmidt subgroups were studied in [4]–[5], and groups with Hall Schmidt
subgroups were described in [6].
A subgroup A of a group G is semi-normal in G if there exists a sub-
group B of G such that G = AB and AB1 is a proper subgroup of G for
every proper subgroup B1 of B. Obviously, every subgroup of prime index
is semi-normal. Also a quasi-normal subgroup (i. e. a subgroup of G, that
permutes with all subgroups of G) is semi-normal. In the simple group
PSL(2, 5) a subgroup A which is isomorphic to the alternating group A4
is a semi-normal Schmidt subgroup, but A is not quasi-normal and not
subnormal.
Some properties of semi-normal subgroups were obtained in [7]–[9].
The criteria for solubility of a group with some semi-normal Schmidt
subgroups were established in [10].
We introduce the following concept, which combines subnormality and
semi-normality.
Definition. A subgroup A of a group G is called semi-subnormal in
G if A is either subnormal in G or semi-normal in G.
In this paper, we establish that a group G with semi-subnormal Schmidt
{2, 3}- subgroups is 3-soluble. Moreover, if all 5-closed Schmidt {2, 5}-
subgroups are semi-subnormal in G, then G is soluble. We prove that
a group with semi-subnormal Schmidt subgroups is metanilpotent.
2. Preliminary results
The terminology in the article as in [11]–[12]. We write Y 6 X if Y is
a subgroup of a group X. Recall that AG = 〈Ag | g ∈ G〉 is the subgroup
generated by all subgroups of G that are conjugate to A. A group with
a normal Sylow p-subgroup is called p-closed. Let π be a set of primes.
We say G is a π-group if every prime divisor of |G| lies in π. Let us agree
to call the S〈p,q〉-group a Schmidt group with a normal Sylow p-subgroup
and a cyclic Sylow q-subgroup.
Lemma 1 ([11, 2.41; 2.43; 5.31]). Let H be a subnormal subgroup of
a group G.
(1) If U 6 G, then U ∩H is subnormal in U . In particular, if H 6
V 6 G, then H is subnormal in V .
(2) If N is a normal subgroup of G, then HN/N is a subnormal
subgroup of G/N .
“adm-n1” — 2020/5/14 — 19:35 — page 68 — #76
68 Finite groups with semi-subnormal Schmidt. . .
(3) If K is a subnormal subgroup of G, then H ∩K and 〈H ∪K〉 are
subnormal subgroups of G.
(4) π(H) = π(HG).
(5) If X is a Fitting class and H ∈ X, then HG ∈ X.
Lemma 2. If X is a subnormal S〈p,q〉-subgroup of a group G, then XG
is a p-closed {p, q}-subgroup.
Proof. Lemma 1 (4) implies that XG is a {p, q}-subgroup. The class of
all p-closed subgroups is a Fitting class [11, 5.3]. By Lemma 1 (5), XG is
p-closed.
Lemma 3 ([4, Lemma 2]). If K and D are subgroups of a group G,
D is normal in K and K/D is an S〈p,q〉-subgroup, then each minimal
supplement L to the subgroup 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 contains an S〈p,q〉-subgroup [P ]Q such that Q is not contained
in D, and L = ([P ]Q)L = QL.
Lemma 4 ([13, Lemma 1.6]). If H is a subgroup of a soluble group G
and |G : H| is a prime number, then G/HG is supersoluble.
Here HG = ∩x∈GH
x is the core of H in G.
The Frattini and Fitting subgroups of a group G are denoted by Φ(G)
and F (G) respectively. The intersection of all normal subgroups of G,
whose quotient groups belong to F is called F-residual of G and denoted
by GF.
Lemma 5 ([12, 24.2; 24.3]). If F is a saturated formation, G is a soluble
minimal non-F-group, then the following statements hold:
(1) GF is a p-group for some p ∈ π(G);
(2) GF/Φ(GF) is a chief factor of G;
(3) GFΦ(G) = F (G).
Lemma 6. If G is a soluble minimal non-metanilpotent group, then
G/F (G) is a Schmidt group.
Proof. The class of all metanilpotent groups is a saturated hereditary
formation [12, p. 36] and coincides with the product NN = N2. First,
let Φ(G) = 1. According to Lemma 5(3), GN2
= F (G). Since GN2
is
a p-group for some p ∈ π(G) by Lemma 5 (1), we have F (G) = Op(G).
Since G /∈ N2, it follows that G/F (G) /∈ N.
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V. N. Kniahina, V. S. Monakhov 69
Let U/F (G) be a proper subgroup of G/F (G). Then U is metanilpotent
andU/F (U) is nilpotent. Since G is soluble, we obtain Op(G) = F (G) 6 U ,
it implies
CG(F (G)) = Z(F (G)), Op′(U) = 1, Op(G) = F (G) 6 F (U) = Op(U).
Let H/F (G) = F (G/F (G)). Then H/F (G) is a p′-subgroup and
H ∩Op(U) = Op(G) = F (G).
Since G/H is nilpotent, UH/H ≃ U/U ∩H and U/Op(U) are nilpotent.
So
U/U ∩H ∩Op(U) = U/F (G)
is nilpotent. Therefore, all proper subgroups of G/F (G) are nilpotent
and G/F (G) is a Schmidt group.
Let Φ(G) 6= 1. According to Lemma 5 (3), the subgroup F (G) =
GN2
Φ(G). Since N2 is a hereditary formation, we have G/Φ(G) /∈ N2 and
G/Φ(G) is a minimal non-N2-group. By the hypothesis, G is soluble, so
F (G/Φ(G)) = F (G)/Φ(G) by [11, 4.21], and
G/F (G) ≃ (G/Φ(G))/(F (G)/Φ(G)) = (G/Φ(G))/(F (G/Φ(G)).
Since Φ(G/Φ(G)) = 1, by the above (G/Φ(G))/F (G/Φ(G)) is a Schmidt
group, therefore G/F (G) is a Schmidt group.
Example 1. In the simple group PSL(2, 5), all its proper subgroups
are metanilpotent, they are even metabelian. Therefore, the condition of
solubility of a group in Lemma 6 is not superfluous.
3. Properties of semi-subnormal subgroups
Lemma 7. (1) If H is a semi-subnormal subgroup of a group G and
H 6 X 6 G, then H is semi-subnormal in X.
(2) If H is a semi-subnormal subgroup of a group G, N is a normal
subgroup of G, then HN is semi-subnormal in G, and HN/N is semi-
subnormal in G/N .
(3) If H is a semi-subnormal subgroup of a group G and Y is nonempty
set of elements in G, then the subgroup
HY = 〈Hy | y ∈ Y 〉
is semi-subnormal in G. In particular, Hg is semi-subnormal in G for any
g ∈ G.
“adm-n1” — 2020/5/14 — 19:35 — page 70 — #78
70 Finite groups with semi-subnormal Schmidt. . .
Proof. Suppose that H is a subnormal subgroup of a group G. Then it
follows from Lemma 1 that statements (1)-(3) are true. If H is semi-normal
in G, then assertions (1)-(3) are proved in [10, Lemma 2.5].
Proposition 1. Let A be a semi-subnormal subgroup of a group G. Then
the following statements holds.
(1) If A is a 2-nilpotent subgroup, then AG is soluble.
(2) If A is soluble and 3 does not divide the order of A, then AG is
soluble.
(3) Let p be the smallest prime divisor of the order of G. If p does not
divide the order of A, then p does not divide the order of AG. In particular,
if A is a subgroup of odd order, then AG is of odd order.
Proof. If A is a semi-normal subgroup of G, then all three statements are
known [10, lemmas 10 and 11]. If A is a subnormal subgroup of G, then
all three statements follow from Lemma 1 (4-5).
Proposition 2. If A is a semi-subnormal subgroup of a soluble group G
and r is the largest prime from π(G), then Or(A) 6 Or(G).
Proof. It suffices to prove that Or(A) is subnormal in G. If A is a subnor-
mal subgroup of G, then Or(A) is subnormal in G and Or(A) 6 Or(G) by
Lemma 1 (5). Let A be a semi-normal subgroup of G and Y be a minimal
supplement to A in G such that A is permutable with all subgroups of Y .
We apply induction on the order of the group. Since Y is soluble, there
is a subgroup X of prime index i.e |Y : X| = t is a prime number. By
the hypothesis, A is permutable with X. Besides A is a semi-subnormal
subgroup of AX by Lemma 7 (1). By induction, Or(A) is subnormal in AX.
If AX is normal in G, then Or(A) is subnormal in G. Suppose that AX
is not normal in G, i. e. AX 6= (AX)G. Since G = AY and AX < G, we
have
1 6= |G : AX| =
|A||Y |
|A ∩ Y |
:
|A||X|
|A ∩X|
=
|Y : X|
|A ∩ Y : A ∩X|
= t.
By Lemma 4, G/(AX)G is supersoluble and is isomorphic to a subgroup
of the symmetric group St of degree t. Since
t = |G : AX| = |G/(AX)G : A/(AX)G|,
then t does not divide |AX/(AX)G|. If t = r, then AX/(AX)G is a r′-
group. Hence
Or(A) 6 Or(AX) 6 (AX)r 6 (AX)G
“adm-n1” — 2020/5/14 — 19:35 — page 71 — #79
V. N. Kniahina, V. S. Monakhov 71
and Or(A) is subnormal in G. If t 6= r, then t < r and G/(AX)G is
a r′-group. Therefore, Ar 6 (AX)G and again Or(A) subnormal in G.
Consequently, Or(A) 6 Or(G).
4. Semi-subnormal Schmidt subgroups
Lemma 8. Suppose that all S〈p,q〉-subgroups of a group G are semi-
subnormal in G.
(1) If H is a subgroup of G, then all S〈p,q〉- subgroups of H are semi-
subnormal in H.
(2) If N is a normal subgroup of G, then all S〈p,q〉- subgroups of G/N
are semi-subnormal in G/N .
Proof. 1. The statement follows from Lemma 7 (1).
2. Let S/N be an S〈p,q〉-subgroup of G/N and L be a minimal subgroup
of S such that S = LN . By Lemma 3, L contains an S〈p,q〉-subgroup A
such that L = AL. By the hypothesis, A is semi-subnormal in G, and
by Lemma 7 (3), L is semi-subnormal in G. Now by Lemma 7 (2), the
subgroup LN/N = S/N is semi-subnormal in G/N .
Theorem 1. Let A be a semi-subnormal Schmidt subgroup of a group G.
(1) If AG is insoluble, then A/Z(A) ≃ A4.
(2) If AG is a simple group, then AG ≃ PSL(2, 5).
Proof. If A is a subnormal subgroup of G, then π(AG) = π(A) by Lemma 2,
so AG is soluble. Further, we assume that A is a semi-normal subgroup
of G. If AG is insoluble, then A is a 2-closed {2, 3}-subgroup according
to [10, Theorem 1]. We deduce from the properties of Schmidt groups [3,
Theorem 1.2] that A/Z(A) ≃ A4. Let AG be a simple group. By Lemma 7,
the subgroup A is semi-normal in AG and |AG : A| is a prime number
by [10, Lemma 7]. Now Z(A) = 1 by [14, V.7.2] and A ≃ A4. In view of
[[15], Theorem 1], we obtain that AG ≃ PSL(2, 5).
Theorem 2. If all {2, 3}-subgroups of a group G are semi-subnormal in
G, then G is 3-soluble.
Proof. We use induction on the order of G. Let N be a normal subgroup
of G. By Lemma 8, all Schmidt {2, 3}-subgroups are semi-subnormal
in N and in G/N . If G 6= N 6= 1, then by induction, N and G/N are
3-soluble. This implies that G likewise is 3-soluble. We will therefore
assume that G is simple, in particular there are no subnormal Schmidt
“adm-n1” — 2020/5/14 — 19:35 — page 72 — #80
72 Finite groups with semi-subnormal Schmidt. . .
subgroups. Consequently, all Schmidt {2, 3}-subgroups are semi-normal
in G. According to [10, Theorem 2], G is 3-soluble.
Corollary 1. If all Schmidt {2, 3}-subgroups and all 5-closed Schmidt
{2, 5}-subgroups of a group G are semi-subnormal, then G is soluble.
In Corollary 1, three types of Schmidt subgroups: S〈2,3〉-subgroups,
S〈3,2〉-subgroups, and S〈5,2〉-subgroups should be semi-subnormal. The
following examples of simple groups show that the condtion of semi-
subnormality of all these types of Schmidt subgroups is not superfluous.
Example 2. In PSL(2, 33), there are no S〈5,2〉-subgroups and S〈3,2〉-
subgroups [16]; so the condition of being semi-subnormal for
S〈2,3〉-subgroups is not superfluous.
Example 3. In SL(2, 8), there are no S〈5,2〉-subgroups and S〈2,3〉-sub-
groups [16]; so the groups with semi-subnormal Schmidt S〈5,2〉-subgroups
and S〈2,3〉-subgroups may be insoluble and the condition of being semi-
subnormal for S〈3,2〉-subgroups is not superfluous.
Example 4. In Sz(8), there are no {2, 3}-subgroups [16]; so the groups
with semi-subnormal Schmidt {2, 3}-subgroups may be insoluble and the
condition of being semi-subnormal for S〈5,2〉-subgroups is not superfluous.
Theorem 3. If all Schmidt subgroups of a group G are semi-subnormal
in G, then G is metanilpotent.
Proof. By Corollary 1, G is soluble. We use induction on the order of G.
According to Lemma 8, in each proper subgroup and in each quotient
group of G different from G, all Schmidt subgroups are semi-subnormal. By
induction, they are metanilpotent. The class of all metanilpotent groups is
a hereditary saturated formation. By [17, Lemma 8], G is primitive, and:
G = [N ]M, M < ·G, N = F (G) = Op(G) = CG(N),
for some p ∈ π(G). According to Lemma 6, the quotient group G/N ≃ M
is a Schmidt group. By the hypothesis, M is either subnormal in G
or semi-normal in G. If M is subnormal in G, then M is normal in G
and M 6 CG(N) = N , a contradiction. If M is semi-normal in G, then |G :
M | = |N | = p is a prime number by [10, Lemma 7]. Hence G is supersoluble
and therefore metanilpotent.
“adm-n1” — 2020/5/14 — 19:35 — page 73 — #81
V. N. Kniahina, V. S. Monakhov 73
References
[1] O. Y. Schmidt, Groups whose all subgroups are special, Matem. Sb., Vol. 31, 1924,
P. 366–372 (in Russian).
[2] N. F. Kuzennyi, S. S. Levishchenko, Finite Shmidt’s groups and their generaliza-
tions, Ukrainian Math. J., Vol. 43 (7-8), 1991, P. 898–904.
[3] V. S. Monakhov, The Schmidt groups, its existence and some applications, Tr.
Ukrain. Math. Congr.–2001. Kiev: 2002, Section 1. P. 81–90 (in Russian).
[4] V. N. Kniahina, V. S. Monakhov, On finite groups with some subnormal Schmidt
subgroups, Sib. Math. J., Vol. 45 (6), 2004, P. 1075–1079.
[5] V. A. Vedernikov, Finite groups with subnormal Schmidt subgroups, Algebra and
Logic, Vol. 46 (6), 2007, P. 363-372.
[6] V. N. Kniahina, V. S. Monakhov, Finite groups with Hall Schmidt subgroups, Publ.
Math. Debrecen, Vol. 81 (3-4), 2012, P. 341–350.
[7] V.V. Podgornaya, Semi-normal subgroups and supersolubility of finite groups,
Vesti Nats.Akad. Navuk Belarusi. Ser. Fiz.-Mat. navuk. Vol. 4, 2000. P. 22–25.
[in Russian]
[8] V. S. Monakhov, Finite groups with a seminormal Hall subgroup, Math. Notes.
Vol. 80 (4), 2006. P. 542–549.
[9] W. Guo, Finite groups with seminormal Sylow subgroups, Acta Math. Sinica.
Vol. 24 (10), 2008. P. 1751–1758.
[10] V.N. Kniahina, V. S. Monakhov, Finite groups with semi-normal Schmidt sub-
groups, Algebra and Logic. Vol. 46 (4), 2007. P. 244–249.
[11] V. S. Monakhov, Introduction to the Theory of Finite groups and their Classes,
Vyshejshaja shkola, 2006 (In Russian).
[12] Shemetkov, L. A. Formations of Finite Groups, M. : Nauka, 1978. [in Russian]
[13] V. S. Monakhov, Finite groups with abnormal and U-subnormal subgroups, Sib.
Math. J. Vol. 57 (2), 2016. P. 353–363.
[14] B. Huppert, Endliche Gruppen I, Springer, 1967.
[15] V. S. Monakhov, Product of a biprimary and a 2-decomposable groups, Math.
Notes, Vol. 23 (5), 1978. P. 355–359.
[16] J.H. Conway, R.T. Curtis, S. P. Norton, R.A. Parker, R.A. Wilson, Atlas of
finite groups, London: Clarendon, 1985.
[17] V. S. Monakhov, Indices of maximal subgroups of finite soluble groups, Algebra
and Logic. Vol. 43 (4), 2004. P. 230–237.
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: 23.04.2019.
V. N. Kniahina, V. S. Monakhov
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