On derived π-length of a finite π-solvable group with supersolvable π-Hall subgroup
It is proved that if π-Hall subgroup is a supersolvable group then the derived π-length of a π-solvable group G is at most 1 + maxr∈π lαr(G), where lαr(G) is the derived r-length of a π-solvable group G.
Saved in:
| Published in: | Algebra and Discrete Mathematics |
|---|---|
| Date: | 2013 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут прикладної математики і механіки НАН України
2013
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/152351 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On derived π-length of a finite π-solvable group with supersolvable π-Hall subgroup / V.S. Monakhov, D.V. Gritsuk // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 233–241. — Бібліогр.: 16 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-152351 |
|---|---|
| record_format |
dspace |
| spelling |
Monakhov, V.S. Gritsuk, D.V. 2019-06-10T11:07:57Z 2019-06-10T11:07:57Z 2013 On derived π-length of a finite π-solvable group with supersolvable π-Hall subgroup / V.S. Monakhov, D.V. Gritsuk // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 233–241. — Бібліогр.: 16 назв. — англ. 1726-3255 2010 MSC:20D10, 20D20, 20F16. https://nasplib.isofts.kiev.ua/handle/123456789/152351 It is proved that if π-Hall subgroup is a supersolvable group then the derived π-length of a π-solvable group G is at most 1 + maxr∈π lαr(G), where lαr(G) is the derived r-length of a π-solvable group G. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On derived π-length of a finite π-solvable group with supersolvable π-Hall subgroup Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
On derived π-length of a finite π-solvable group with supersolvable π-Hall subgroup |
| spellingShingle |
On derived π-length of a finite π-solvable group with supersolvable π-Hall subgroup Monakhov, V.S. Gritsuk, D.V. |
| title_short |
On derived π-length of a finite π-solvable group with supersolvable π-Hall subgroup |
| title_full |
On derived π-length of a finite π-solvable group with supersolvable π-Hall subgroup |
| title_fullStr |
On derived π-length of a finite π-solvable group with supersolvable π-Hall subgroup |
| title_full_unstemmed |
On derived π-length of a finite π-solvable group with supersolvable π-Hall subgroup |
| title_sort |
on derived π-length of a finite π-solvable group with supersolvable π-hall subgroup |
| author |
Monakhov, V.S. Gritsuk, D.V. |
| author_facet |
Monakhov, V.S. Gritsuk, D.V. |
| publishDate |
2013 |
| language |
English |
| container_title |
Algebra and Discrete Mathematics |
| publisher |
Інститут прикладної математики і механіки НАН України |
| format |
Article |
| description |
It is proved that if π-Hall subgroup is a supersolvable group then the derived π-length of a π-solvable group G is at most 1 + maxr∈π lαr(G), where lαr(G) is the derived r-length of a π-solvable group G.
|
| issn |
1726-3255 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/152351 |
| citation_txt |
On derived π-length of a finite π-solvable group with supersolvable π-Hall subgroup / V.S. Monakhov, D.V. Gritsuk // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 233–241. — Бібліогр.: 16 назв. — англ. |
| work_keys_str_mv |
AT monakhovvs onderivedπlengthofafiniteπsolvablegroupwithsupersolvableπhallsubgroup AT gritsukdv onderivedπlengthofafiniteπsolvablegroupwithsupersolvableπhallsubgroup |
| first_indexed |
2025-11-27T01:39:22Z |
| last_indexed |
2025-11-27T01:39:22Z |
| _version_ |
1850791453573251072 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 16 (2013). Number 2. pp. 233 – 241
c© Journal “Algebra and Discrete Mathematics”
On derived π-length of a finite π-solvable group
with supersolvable π-Hall subgroup
V. S. Monakhov, D. V. Gritsuk
Communicated by L. A. Kurdachenko
To memory of L. A. Shemetkov
Abstract. It is proved that if π-Hall subgroup is a super-
solvable group then the derived π-length of a π-solvable group G is
at most 1 + maxr∈π la
r
(G), where la
r
(G) is the derived r-length of a
π-solvable group G.
Introduction
All groups considered in this paper will be finite. All notation and
definitions correspond to [1], [2].
Let P be the set of all prime numbers, and let π be the set of primes.
In this paper, π′ is the set of all primes not contained in π, i. e. π = P \ π′.
By π also denotes a function defined on the set of natural numbers N
as follows: π(a) is the set of primes dividing a positive integer a. For
a group G and a subgroup H of G we believe that π(G) = π(|G|) and
π(G : H) = π(|G : H|).
Fix a set of prime numbers π. If π(m) ⊆ π, then a positive integer m
is called a π-number. The group G is called a π-group if π(G) ⊆ π, and a
π′-group if π(G) ⊆ π′. If G is a π′-group, then π(G) ∩ π = ∅. The chain
of subgroups
G = G0 ⊇ G1 ⊇ G2 ⊇ . . . ⊇ Gn−1 ⊇ Gn = 1, (1)
2010 MSC: 20D10, 20D20, 20F16.
Key words and phrases: finite group, π-soluble group, supersolvable group,
π-Hall subgroup, derived π-length.
234 On derived π-length of a finite π-solvable group.. .
is called subnormal series of a group G, if subgroup Gi+1 is normal
subgroup of Gi for every i. The quotient groups Gi/Gi+1 are called
factors of the series (1).
The group is called a π-solvable group if it has a subnormal series
(1) whose factors are solvable π-groups or π′-groups. The least number
of π-factors of all such subnormal series of a group G is called the π-
length of a π-solvable group G and is denoted by lπ(G). Since π-factors
of subnormal series (1) of a π-solvable group G are solvable, then every
π-solvable group has subnormal series in which all π-factors are nilpotent.
The least number of nilpotent π-factors of all such subnormal series of
a group G is called the nilpotent π-length of a π-solvable group G and
is denoted by lnπ(G). In case when π consists of a single prime {p}, i. e.
π = {p}, we will obtain lπ(G) = lnπ(G) = lp(G) for every π-solvable group.
The equality lπ(G) = lnπ(G) is cleared to be a true for a π-solvable group
with nilpotent π-Hall subgroup.
Recall that least positive integer m such that G(m) = 1 is called the
derived length of the group G and is denoted by d(G). Here G′ is the
derived subgroup of G and G(i) = (G(i−1))′.
V. S. Monakhov suggested a new notation of the derived π-length of a
π-solvable group. Let G be a π-solvable group. Then G has a subnormal
series (1) whose factors are π′-groups or abelian π-groups. The least
number of abelian π-factors of all such subnormal series of a group G is
called the derived π-length of a π-solvable group G and is denoted by
laπ(G). Clearly, in the case π = π(G) to laπ(G) coincides with the derived
length of G. The initial properties of the derived π-length and some of
the results related to this notion, established in [4] – [6].
In 2001 V. S. Monakhov and O. A. Shpyrko [3] proved that lnπ(G) ≤
1 + maxr∈π lr(G) if G is a π-solvable group in which the derived subgroup
of a π-Hall subgroup is nilpotent. In this article, we received an analogue
of this result for the derived π-length. Also, we obtain a new estimate of
derived π-length of a π-solvable group whose all proper subgroups of a
π-Hall subgroup are supersolvable.
1. Preliminary results
Lemma 1 ([4, Lemma 3]). Let G be a π-solvable group. Then d(Gπ) ≤
laπ(G) ≤ lπ(G)d(Gπ).
Here and below, Gπ is a π-Hall subgroup of a π-solvable group G.
V. S. Monakhov, D. V. Gritsuk 235
Lemma 2 ([4, Lemma 4]). Let G be a π-solvable group, and let t be a
positive integer. Suppose that laπ(G/N) ≤ t for every non-trivial subgroup
N of G, but laπ(G) > t. Then:
(1) Oπ′(G) = 1;
(2) G has a unique minimal normal subgroup;
(3) F (G) = Op(G) = F (Oπ(G)) for some prime p ∈ π;
(4) Op′(G) = 1 and CG(F (G)) ⊆ F (G).
Here F (X) is the Fitting subgroup of a group X, i. e. F (X) is the
largest normal nilpotent subgroup of X.
Lemma 3 ([4, Theorem 1]). If G is a π-solvable group in which a Sylow
p-subgroup is abelian for every p ∈ π, then laπ(G) = d(Gπ) ≤ |π(Gπ)|.
Lemma 4 ([4, Theorem 2]). Let G be a π-solvable group. If Gπ is abelian,
then laπ(G) ≤ 1.
Lemma 5 ([5, Lemma 2.6]). If G is a π-solvable group and π = π1 ∪ π2,
then laπ(G) ≤ laπ1
(G) + laπ2
(G).
Lemma 6 ([5, Theorem 3.1]). Let G be a p-solvable group. If a Sylow
p-subgroup of G is bicyclic, then lap(G) ≤ 2 for every p > 2 and lap(G) ≤ 3
for p = 2.
The group is called a bicyclic group if it is the product of two cyclic
subgroups.
Lemma 7 ([7, Theorem 2]). Let G be a group of odd order. If every Sylow
subgroup of G is bicyclic, then the derived subgroup of G is nilpotent.
A group is called a Schmidt group if it is a non-nilpotent groups all
of whose proper subgroups are nilpotent. O. Yu. Schmidt pioneered the
study of such groups [8]. A whole paragraph from Huppert’s monography
is dedicated to Schmidt groups, (see [1, III.5]). A survey of results on
the existence of Schmidt subgroups in finite groups and some of their
applications in the theory of group classes given in [9].
Lemma 8 ([10, Theorem 2]). Let G be a p-solvable group. If a Sylow
p-subgroup of G is isomorphic to a Sylow Subgroup of a Schmidt group,
then lap(G) ≤ 1.
The group is called a Miller-Moreno group if it is a non-abelian group
and all of its proper subgroups are abelian. Non-nilpotent Miller-Moreno
groups are a special case of Schmidt groups and the structure of these
236 On derived π-length of a finite π-solvable group.. .
groups is easily derived from the properties of Schmidt groups. Nilpotent
Miller-Moreno groups are the groups of prime-power order.
We denote by U a class of all supersolvable groups. Then GU is
U-residual of G, i. e. GU is the intersection of all those normal subgroups
N of G for which G/N ∈ U.
Lemma 9 ([11, Theorem 22], [12]). Let G be a minimal non-supersolvable
group, i. e. G is a non-supersolvable group and all proper subgroups of G
are supersolvable. Then:
(1) G is solvable and |π(G)| ≤ 3;
(2) GU = P is a Sylow p-subgroup of G and P/Φ(P ) is a minimal
normal subgroup of G/Φ(G);
(3) P ′ ⊆ Φ(P ) ⊆ Z(P );
(4) if Q is a complement to P in G, then Q/Q ∩ Φ(G) is either a
cyclic group of prime-power order or a Miller-Moreno group.
Lemma 10 ([13, Theorem 26.3], [14, Theorem 1]). The minimal non-
supersolvable groups are one of the following types:
(1) G = [P ]Q is a Schmidt group;
(2) G = [P ]Q, where P is a Sylow p-subgroup of Schmidt type (see the
definition in [14]); Q is a cyclic Sylow q-subgroup; [P ]Φ(Q) and [Φ(P )]Q
are supersolvable, [P, Φ(Q)] = P ;
(3) G = [P ]Q, where P is a Sylow p-subgroup of Schmidt type; Q is a
Sylow q-subgroup; Φ(Q) > CQ(P ) ⊳ G; Q/CQ(P ) is either a non-abelian
group of order q3 and exponent q or a Miller-Moreno group of prime-power
order containing a cyclic maximal subgroup, p ≡ 1 (mod q); [P, Q′] = P,
[Φ(P )]Q, [P ]Q1 are supersolvable, where Q1 is any subgroup of Q;
(4) G = [P ]([Q]R), where P is a Sylow p-subgroup of Schmidt type;
Q and R are the cyclic Sylow q- and r-subgroups, q > r; [P ]Q, [P ]R and
[Q]R are non-nilpotent; [P, Q] = P ; [Q, R] = Q; Φ(P ) < Φ(P )·[P, R] ≤ P ;
Φ(P ) × Φ(Q) ≤ Z([P ]Q); Φ(R) = Z([Q]R), p ≡ 1 (mod qr) and q ≡ 1
(mod r).
Lemma 11. Let G be a π-solvable group, and let Gπ be a minimal non-
supersolvable group. Then lp(G) ≤ 1 and lap(G) ≤ 2 for p ∈ π((Gπ)U).
Proof. By hypothesis, (Gπ)U = Gp. First of all, we prove that lp(G) ≤ 1.
The group GπOp′(G)/Op′(G) is a π-Hall subgroup of G/Op′(G) and
(GπOp′(G)/Op′(G))U = (Gπ)UOp′(G)/Op′(G) ≃
≃ GpOp′(G)/Op′(G) ≃ Gp ≃ (Gπ)U
V. S. Monakhov, D. V. Gritsuk 237
by properties residuals. The group GπOp′(G)/Op′(G) is a minimal non-
supersolvable group and, by induction, lp(G/Op′(G)) ≤ 1, so lp(G) ≤ 1.
Hence we can assume that Op′(G) = 1. Therefore, F (G) = Op(G) and
CG(Op(G)) ⊆ Op(G).
Assume that Op(G) is a proper subgroup of Gp. Clearly, the group
Op(G)Φ(Gp)/Φ(Gp) is a normal subgroup of Gπ/Φ(Gp). Since Gp/Φ(Gp)
is a minimal normal subgroup of Gπ/Φ(Gp) by Lemma 9 (2) and
Op(G)Φ(Gp)/Φ(Gp) ⊆ Gp/Φ(Gp),
then
Op(G)Φ(Gp)/Φ(Gp) = 1 or Op(G)Φ(Gp) = Gp.
If Op(G)Φ(Gp)/Φ(Gp) = 1, then Op(G) ⊆ Φ(Gp). Since, by Lemma 9 (3),
Φ(Gp) ⊆ Z(Gp), we have
Op(G) ⊆ Z(Gp), Gp ⊆ CG(Op(G)) ⊆ Op(G),
we have a contradiction. If Op(G)Φ(Gp) = Gp, then Op(G) = Gp. There-
fore, Op(G) = Gp. Hence lp(G) ≤ 1.
By Lemma 9 (3), d(Gp) ≤ 2, and lap(G) ≤ 2 by Lemma 1.
2. Main results
Theorem 1. Let G be a π-solvable group. If the derived subgroup of Gπ
is nilpotent, then laπ(G) ≤ 1 + maxr∈π lar (G).
Proof. Let G be a π-solvable group, and let the derived subgroup of Gπ
be a nilpotent. We use induction on |G|. Let N is a normal subgroup
of G. Since GπN/N ≃ Gπ/(Gπ ∩ N), then their derived subgroups are
isomorphic.
(Gπ/(Gπ ∩ N))′ = (Gπ)′(Gπ ∩ N)/(Gπ ∩ N) ≃
≃ (Gπ)′/((Gπ)′ ∩ N) ≃ (GπN/N)′.
Therefore, the conditions of the lemma are inherited by all quotient groups.
By Lemma 2, Oπ′(G) = 1, G has a unique minimal normal subgroup
CG(F (G)) ⊆ F (G) = Op(G) = F (Oπ(G))
for some prime p ∈ π. Clearly, F (G) ⊆ Gπ.
238 On derived π-length of a finite π-solvable group.. .
Let K be the derived subgroup of Gπ. By hypothesis of the theorem
subgroup K is nilpotent. Since p′-Hall subgroup Kp′ of K is a normal
subgroup of Gπ, it follows
Kp′ ⊆ CG(F (G)) ⊆ F (G), Kp′ = 1.
Thus, K is a p-group, Gπ\{p} is abelian and a Sylow q-subgroup of G
is abelian for every q ∈ π\{p}. So laq (G) = 1 for every q ∈ π\{p} by
Lemma 4. Therefore, maxr∈π lar (G) = lap(G).
Let π1 = π \ {p}. By Lemma 5, laπ(G) ≤ laπ1
(G) + lap(G). Since Gπ1
is abelian, we have laπ1
(G) ≤ 1 by Lemma 4. Now laπ(G) ≤ 1 + lap(G) ≤
1 + maxr∈π lar (G).
Corollary 1. Let G be a π-solvable group. If a Sylow p-subgroup of G is
cyclic for every p ∈ π, then laπ(G) ≤ 2.
Proof. By Lemma 4, lap(G) ≤ 1 for all p ∈ π, so maxr∈π lar (G) ≤ 1 and,
by [1, Theorem IV.2.11], Gπ is a supersolvable. By [1, Theorem VI.9.1],
the derived subgroup of Gπ is nilpotent. By Theorem 1, laπ(G) ≤ 2.
Corollary 2. Let G be a π-solvable group, and let a Sylow p-subgroup of
G be bicyclic for every p ∈ π. Then laπ(G) ≤ 6. If 2 /∈ π, then laπ(G) ≤ 3.
Proof. Let π = {2}∪τ. By Lemma 5, laπ(G) ≤ la2(G)+ laτ (G). By Lemma 6,
la2(G) ≤ 3 and lat (G) ≤ 2 for all t ∈ τ, so maxt∈τ lat (G) ≤ 2. By Lemma 7,
the derived subgroup of a τ -Hall subgroup is nilpotent. By Theorem 1,
we have that
laτ (G) ≤ 1 + max
t∈τ
lat (G) ≤ 3.
Now laπ(G) ≤ 6. If 2 /∈ π, then π = τ and laπ(G) = laτ (G) ≤ 3.
Let H be a subgroup of a group G. A subgroup K of G is called a
complement of H in G if G = HK and H ∩K = 1. Yu. M. Gorchakov [15]
showed that complementability of all subgroups is equivalents to comple-
mentability subgroups of prime order. The group G is called completely
factorable if all of its subgroups are complemented. In 1937 Ph. Hall
[16] found that finite groups in which all subgroups are complemented
exhausted by supersolvable groups with elementary abelian Sylow subgroups.
Corollary 3. Let G be a π-solvable group. If Gπ is completely factorable,
then laπ(G) ≤ 2.
V. S. Monakhov, D. V. Gritsuk 239
Proof. By [16], Gπ of G is supersolvable and a Sylow p-subgroup of G is
an elementary abelian for all p ∈ π. By [1, Theorem VI.9.1], the derived
subgroup of Gπ is nilpotent. By Lemma 4 and Theorem 1, laπ(G) ≤ 2.
Corollary 4. Let G be a π-solvable group. If Gπ is supersolvable, then
laπ(G) ≤ 1 + maxr∈π lar (G).
Proof. By [1, Theorem VI.9.1], the derived subgroup of Gπ is nilpotent.
By Theorem 1, laπ(G) ≤ 1 + maxr∈π lar (G).
Corollary 5. Let G be a π-solvable group. If Gπ is a Schmidt group,
then laπ(G) ≤ 3.
Proof. Let G be a π-solvable group, and let Gπ = [P ]Q be a Schmidt
group, when P is a normal Sylow p-subgroup, and Q is a non-normal
Sylow q-subgroup. Since Q is cyclic, we have laq (G) ≤ 1 by Lemma 4.
By Lemma 8, lp(G) ≤ 1. Since either P is abelian or P ′ = Z(P ) [8]–[9],
we have d(P ) ≤ 2. By Lemma 1, lap(G) ≤ 2. By Lemma 5, laπ(G) ≤
lap(G) + laq (G) ≤ 3.
Corollary 6. Let G be a π-solvable group. If Gπ is a Miller-Moreno
group, then laπ(G) ≤ 2.
Proof. Assume that Gπ is not a group of prime-power order. Then Gπ
is a Schmidt group in which every Sylow subgroup is abelian. So the
derived subgroup of Gπ is abelian and maxr∈π lar (G) ≤ 1 by Lemma 3.
By Theorem 1, laπ(G) ≤ 2.
Let Gπ = Gp be a group of prime-power order. We use induction
on |G|. If N is a non-trivial normal subgroup of G, then GpN/N is an
abelian or a Miller-Moreno group. So lap(G/N) ≤ 2 either by Lemma 4 or
by induction. By Lemma 2, G has a unique minimal normal subgroup,
Op′(G) = 1, F (G) = Op(G), CG(F (G)) ⊆ F (G).
If F (G) = Gp, then lap(G) = d(Gp) = 2. Let F (G) be a proper subgroup
of Gp. Then F (G) ⊆ M, when M is some maximal subgroup of Gp.
By condition, M is abelian. So M ⊆ CG(F (G)) and F (G) = M. Now
Gp/F (G) has prime order and lap(G/F (G)) ≤ 1 by Lemma 4. Since F (G)
is abelian, we have lap(G) ≤ 2.
Theorem 2. Let G be a π-solvable group. If every proper subgroup of
Gπ is supersolvable, then laπ(G) ≤ 2 + maxr∈π lar (G).
240 On derived π-length of a finite π-solvable group.. .
Proof. If Gπ is supersolvable, then laπ(G) ≤ 1 + maxr∈π lar (G) by Corol-
lary 4. Let Gπ be a non-supersolvable group. Then Gπ is one of the four
types listed in Lemma 10. Notation for Gπ corresponds to Lemma 10. By
Lemma 11, lap(G) ≤ 2.
If Gπ is a group of type (1)–(2), then Q is cyclic and laq (G) ≤ 1 by
Lemma 4 and, by Lemma 5,
laπ(G) ≤ lap(G) + laq (G) ≤ 2 + 1 ≤ 2 + max
r∈π
lar (G).
If Gπ is a group of type (3), then, by Lemma 5,
laπ(G) ≤ lap(G) + laq (G) ≤ 2 + laq (G) ≤ 2 + max
r∈π
lar (G).
Let Gπ be a group of type (4). Then Gπ = [P ]([Q]R), where Q and
R are cyclic Sylow q- and r-subgroups. By Lemma 5, laπ(G) ≤ la{p,q}(G) +
lar (G). Since {p, q}-Hall subgroup of group G is supersolvable, we have
la{p,q}(G) ≤ 1 + maxt∈{p,q} lat (G) by Corollary 4. By Lemma 4, lar (G) ≤ 1,
and by Lemma 5,
laπ(G) ≤ la{p,q}(G) + lar (G) ≤ 1 + max
t∈{p,q}
lat (G) + 1 ≤ 2 + max
t∈π
lat (G).
References
[1] B. Huppert, Endliche Gruppen, I. Berlin–Heidelberg. New York: Springer–Verlag,
1967.
[2] V. S. Monakhov, Introduction to the theory of finite groups and their classes,
Minsk: Higher School, 2006 (in Russian).
[3] V. S. Monakhov, O. A. Shpyrko, On nilpotent π-length of a finite π-solvable group,
Discrete Mathematics, V. 13, N. 3, 2001. pp. 145-152 (in Russian).
[4] D. V. Gritsuk, V. S. Monakhov, O. A. Shpyrko, On derived π-length of a π-solvable
group, BSU Vestnik, Series 1. N. 3, 2012. pp. 90-95 (in Russian).
[5] D. V. Gritsuk, V. S. Monakhov, O. A. Shpyrko, On finite π-solvable groups with bi-
cyclic Sylow subgroups, Promlems of Physics, Mathematics and Technics, N. 1(14),
2013, pp. 61-66 (in Russian).
[6] D. V. Gritsuk, V. S. Monakhov, On solvable groups whose Sylow subgroups are
either abelian or extraspecial, Proceedings of the Institute of Mathematics of NAS
of Belarus, Volume 20, N. 2, 2012. pp. 3-9 (in Russian).
[7] V. S. Monakhov, E. E. Gribovskaya, On maximal and Sylow subgroups of a finite
solvable groups, Mathematical Notes, Volume 70, N. 4, 2001. pp. 603-612 (in
Russian).
[8] O. Yu. Schmidt, Groups whose all subgroups are special, Mathematics Sbornik,
Volume 31, 1924, pp. 366-372 (in Russian).
V. S. Monakhov, D. V. Gritsuk 241
[9] V. S. Monakhov, The Schmidt subgroups, its existence, and some of their classes,
Volume Section 1, Tr. Ukraini. Mat. Congr., 2001, Kiev, 2002, pp. 81-90 (in
Russian).
[10] L. A. Shemetkov, Yi. Xiaolan, On the p-length of finite p-soluble groups, Proceedings
of the Institute of Mathematics of NAS of Belarus, Volume 16, N. 1, 2008, pp.
93-96.
[11] B. Huppert, Normalteiler und maximale Untergruppen endlicher Gruppen, Mathe-
matische Zeitschrift, Bd. 60, 1954, pp. 409-434.
[12] K. Doerk, Minimal nicht überauflösbare, endliche Gruppen, Mathematische
Zeitschrift, Bd. 91. 1966, pp. 198-205.
[13] V. T. Nagrebetskii, On finite minimal non-supersolvable groups, Finite groups,
Minsk: Science and Technics, 1975, pp. 104-108 (in Russian).
[14] S. S. Levischenko, N. Ph. Kuzenny, Constructive description of finite minimal
non-supersolvable groups, Questions in algebra, Minsk, 1987, N. 3, pp.56-63 (in
Russian).
[15] Yu. M. Gorchakov, Primitive factorable groups, Proceedings of the University of
Perm, N. 17, 1960, pp. 15-31 (in Russian).
[16] Ph. Hall, Complemented group, J. London Math. Soc., V. 12, 1937, pp. 201-204.
Contact information
V. S. Monakhov Department of Mathematics, Gomel Francisk
Skorina State University, Gomel, Belarus
E-Mail: Victor.Monakhov@gmail.com
D. V. Gritsuk Department of Mathematics, Gomel Francisk
Skorina State University, Gomel, Belarus
E-Mail: Dmitry.Gritsuk@gmail.com
Received by the editors: 18.05.2013
and in final form 18.05.2013.
|