On minimal ω-composition non-H-formations
Let H be some class of groups. A formation F is called a minimal τ -closed ω-composition non-H-formation [1] if F * H but F1 ⊆ H for all proper τ -closed ω-composition subformations F₁ of F. In this paper we describe the minimal τ -closed ω-composition non-H-formations, where H is a 2-multiply lo...
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Інститут прикладної математики і механіки НАН України
2006
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| Назва видання: | Algebra and Discrete Mathematics |
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| Цитувати: | On minimal ω-composition non-H-formations / L.I. Belous, V.M. Selkin // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 4. — С. 1–11. — Бібліогр.: 12 назв. — англ. |
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irk-123456789-1573902019-09-01T01:25:13Z On minimal ω-composition non-H-formations Belous, L.I. Selkin, V.M. Let H be some class of groups. A formation F is called a minimal τ -closed ω-composition non-H-formation [1] if F * H but F1 ⊆ H for all proper τ -closed ω-composition subformations F₁ of F. In this paper we describe the minimal τ -closed ω-composition non-H-formations, where H is a 2-multiply local formation and τ is a subgroup functor such that for any group G all subgroups from τ (G) are subnormal in G. 2006 Article On minimal ω-composition non-H-formations / L.I. Belous, V.M. Selkin // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 4. — С. 1–11. — Бібліогр.: 12 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20D20. http://dspace.nbuv.gov.ua/handle/123456789/157390 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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English |
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Let H be some class of groups. A formation F
is called a minimal τ -closed ω-composition non-H-formation [1] if
F * H but F1 ⊆ H for all proper τ -closed ω-composition subformations F₁ of F. In this paper we describe the minimal τ -closed
ω-composition non-H-formations, where H is a 2-multiply local formation and τ is a subgroup functor such that for any group G all
subgroups from τ (G) are subnormal in G. |
| format |
Article |
| author |
Belous, L.I. Selkin, V.M. |
| spellingShingle |
Belous, L.I. Selkin, V.M. On minimal ω-composition non-H-formations Algebra and Discrete Mathematics |
| author_facet |
Belous, L.I. Selkin, V.M. |
| author_sort |
Belous, L.I. |
| title |
On minimal ω-composition non-H-formations |
| title_short |
On minimal ω-composition non-H-formations |
| title_full |
On minimal ω-composition non-H-formations |
| title_fullStr |
On minimal ω-composition non-H-formations |
| title_full_unstemmed |
On minimal ω-composition non-H-formations |
| title_sort |
on minimal ω-composition non-h-formations |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/157390 |
| citation_txt |
On minimal ω-composition non-H-formations / L.I. Belous, V.M. Selkin // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 4. — С. 1–11. — Бібліогр.: 12 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT belousli onminimalōcompositionnonhformations AT selkinvm onminimalōcompositionnonhformations |
| first_indexed |
2025-07-14T09:49:39Z |
| last_indexed |
2025-07-14T09:49:39Z |
| _version_ |
1837615367614103552 |
| fulltext |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 4. (2006). pp. 1 – 11
c© Journal “Algebra and Discrete Mathematics”
On minimal ω-composition non-H-formations
Liudmila I. Belous, Vadim M. Sel’kin
Communicated by L. A. Shemetkov
Abstract. Let H be some class of groups. A formation F
is called a minimal τ -closed ω-composition non-H-formation [1] if
F * H but F1 ⊆ H for all proper τ -closed ω-composition subfor-
mations F1 of F. In this paper we describe the minimal τ -closed
ω-composition non-H-formations, where H is a 2-multiply local for-
mation and τ is a subgroup functor such that for any group G all
subgroups from τ(G) are subnormal in G.
Introduction
Throughout this paper all groups considered are finite. A non-empty set
of formations Θ is called a full lattice of formations [2] if the intersection
of any set of formations from Θ again belongs to Θ and in Θ there is a
formation F such that H ⊆ F for all H ∈ Θ. Formations belonging to Θ
are called Θ-formations. Let H be some class of groups. Recall that a
Θ-formation F is called a minimal non-H-Θ-formation (L.A. Shemetkov
[1]) or HΘ-critical formation (A.N. Skiba [3]) if F * H but F1 ⊆ H for all
proper Θ-subformations F1 of F.
The minimal non-H-Θ-formations, where Θ is the set of all saturated
formations have been described in work [4]. This result have been applied
in research of local formations with given subformations (see for more in
details Chapter 4 in [5]). In the book [2] analogue of this result in the
class of τ -closed saturated formations have been obtained. In the work
[6] the minimal ω-saturated non-H-formations, where H is a 2-multiply
local formation have been described. In [7] the minimal ω-saturated
2000 Mathematics Subject Classification: 20D20.
Key words and phrases: formation, τ -closed ω-composition, satellite.
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.2 On minimal τ -closed ω-composition non-H-formations
non-H-formations, where H is an any formation of classical type have
been described. In the work [9] the structure of the minimal non-H-Θ-
formations, where Θ is a class of all ω-composition formations has been
described.
In this paper we describe the minimal τ -closed ω-composition non-H-
formations, where H is a 2-multiply local formation and τ is a subgroup
functor such that for any group G all subgroups from τ(G) are subnormal
in G.
1. Preliminaries
We use standard terminology [10], [11]. In addition we shall need some
definitions and notations from the work of L.A.Shemetkov and A.N. Skiba
[8] and the concept of subgroup functor given by A.N.Skiba [2].
Let L be an arbitrary non-empty class of abelian simple groups and
ω = π(L). Every function
f : ω
⋃
{ω′} −→ {formations of groups}
is called an ω-composition satellite.
We use Cp(G) to denote the intersection of all centralizers of abelian
chief p-factors of the group G (we write Cp(G) = G if G has no such
chief factors). Let R(G) denote the radical of G (i.e. R(G) is the largest
normal soluble subgroup of G).
Let X be a set of groups. We use Com(X) to denote the class of
all abelian simple groups A such that A ≃ H/K for some composition
factor H/K of some group G ∈ X. Also, we write Com(G) for the set
Com({G}).
For an arbitrary ω-composition satellite f we put following [8]
CFω(f) = {G | G/(R(G) ∩ Oω(G)) ∈ f(ω′) and G/Cp(G) ∈ f(p) for all
p ∈ π(Com(G)) ∩ ω}.
If the formation F is such that F = CFω(f) for some ω-composition
satellite f , then we say that F is an ω-composition formation and f is an
ω-composition satellite of that formation [8]. A ω-composition satellite f
of a ω-composition formation F is called an inner ω-composition satellite
of F if f(a) ⊆ F for all a ∈ ω ∪ {ω′}.
Recall that a Skiba subgroup functor τ [2] associates with every group
G a system of its subgroups τ(G) such that the following conditions hold:
1) G ∈ τ(G) for any group G;
2) for any epimorphism ϕ : A −→ B and for any groups H ∈ τ(A)
and T ∈ τ(B) we have Hϕ ∈ τ(B) and Tϕ−1
∈ τ(A).
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.L. I. Belous, V. M. Sel’kin 3
We write τ1 ≤ τ2 if and only if τ1(G) ⊆ τ2(G).
If for all groups H and G, where H ∈ τ(G) we have τ(H) ⊆ τ(G),
then they say that τ is a closed subgroup functor.
Let τ be the intersection of all closed functors τi such that τ ≤ τi.
The functor τ is called the closure of τ .
In this paper we consider the only subgroup functors τ such that for
any group G the set τ(G) consists of some subnormal subgroups of G.
A formation F is called τ -closed if τ(G) ⊆ F for any group G ∈ F. A
satellite f is called τ -valued if all values of f are τ -closed formations.
We denote by cτ
ωform(X) the intersection of all τ -closed ω-compo-
sition formations containing the set of groups X. Then cτ
ωform(X) is
called the τ -closed ω-composition formation generated by X. If X =
{G} for some group G, then instead of cτ
ωform(G) we write cτ
ωformG.
Formations of this kind are called one-generated τ -closed ω-composition
formations.
Let {fi | i ∈ I} be the set of ω-composition satellites. Then
⋂
i∈I fi
is a satellite such that (
⋂
i∈I fi)(a) =
⋂
i∈I fi(a) for all a ∈ ω ∪ {ω′}.
Now let {fi | i ∈ I} be the set of all ω-composition τ -valued satellites
of the formation F. By Lemma 2 [8], f =
⋂
i∈I fi is a ω-composition
satellites of F. The satellite f is called the minimal ω-composition τ -
valued satellite of F.
Let f be the minimal ω-composition τ -valued satellite of F. And let
F be a satellite such that
F (a) =
{
Npf(p), if a = p ∈ ω;
F, if a = ω′.
Then F is a ω-composition satellite of the formation F [8] and it is
called the canonical ω-composition satellite of F.
Let f and h be two ω-composition satellites of the formation F. Then
we write f ≤ h if for all a ∈ ω ∪ {ω′} we have f(a) ⊆ h(a).
Lemma 1.1. [8, 1]. Let G be a group, p be a prime. Assume that
N E G and that for every composition factor H/K of the subgroup N we
have p 6= |H/K|. Then Cp(G/N) = Cp(G)/N .
Lemma 1.2. [12, 2]. Let p be a prime, Op(G) = 1 and T = Zp ≀ G =
[K]G, where K is the base group of T . Then K = Cp(T ).
Lemma 1.3. [8, 4]. Let F = CFω(f) and p ∈ ω. If G/Op(G) ∈ F∩f(p),
then G ∈ F.
Lemma 1.4. [8, 5]. Let F be an arbitrary non-empty set of groups and
X ⊆ H, where H is a τ -closed formation. Let F = cτ
ωfrom(X) and π =
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.4 On minimal τ -closed ω-composition non-H-formations
π(Com(X)). Then F has the minimal τ -valued ω-composition satellite f
and f has the following values:
(1) f(ω′) = τ form(G/(Oω(G) ∩ R(G))|G ∈ X).
(2) f(p) = τ form(G/Cp(G)|G ∈ X), for all p ∈ π ∩ ω.
(3) f(p) = ∅, for all p ∈ ω \ π.
(4) If F = CFω(h) and h be the τ -valued satellite, then
f(p) = τ form(A | A ∈ h(p) ∩ F, Op(A) = 1)
for all p ∈ π ∩ ω and
f(ω′) = τ form(A | A ∈ h(ω′) ∩ F and R(A) ∩ Oω(A) = 1).
Lemma 1.5. [8, 6]. Let fi be the minimal ω-composition satellite of the
formation Fi, i = 1, 2. Then F1 ⊆ F2 if and only if f1 ≤ f2.
Lemma 1.6. [2, 2.1.5]. Let A be a monolithic group and R * Φ(A) is
the socle of G. Then the formation F = τformA is a τ -irreducible and
M = τform(X∪{A/R}) is the unique maximal τ -closed subformation of
F, where X is the set of all proper τ -subgroups of A.
2. Main results
A formation F is called a 2-multiply local if it has a local satellite f such
that all non-empty values of f are local formations.
Theorem 2.1. Let f be the minimal τ -valued ω-composition satellite of
the formation F and let H be the canonical ω-composition satellite of a
2-multiply local formation H. A formation F is a minimal τ -closed ω-
composition non-H-formation if and only if F = cτ
ωformG where G is a
monolithic τ -minimal non-H-group and R = GH = Soc(G) is the socle of
G, where R * Φ(G) and either π = π(Com(R)) ∩ ω = ∅ or π 6= ∅ and
G satisfies one of the following conditions:
1) G = R is a group of prime order p /∈ π(H);
2) G = [R]M , where R = Op(G) = Fp(G) for same p ∈ π and M is a
monolithic τ -minimal non-H(p)-group and Q = MH(p) = Soc(M) is the
socle of M , where p 6∈ π(Com(Q)) and Q * Φ(M).
Proof. Necessity. Let G be a group of minimal order in F\H. Then G
is a monolithic τ -minimal non-H-group and R = GH 6= 1 is the socle
of G. Let F 6= cτ
ωformG. Then cτ
ωformG ⊆ H and so G ∈ H. This
contradiction shows that F = cτ
ωformG. Since by hypothesis H is a local
formation, then by Theorem 4.3 [11], H is a saturated formation and so
R * Φ(G).
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.L. I. Belous, V. M. Sel’kin 5
Let π = π(Com(R)) ∩ ω = ∅. In this case the condition of the
theorem is carried out.
Let’s consider the case π 6= ∅ and p ∈ π.
Assume that G = CG(R). Let R 6= G. Assume that H(p) = ∅.
Consequently p /∈ π(H) and so Zp /∈ H. Then Np * H. Consequently
Np = F and G is a p-group. So G/R ∈ H and G/R is a p-group. If
G/R 6= 1, then p ∈ π(H). This contradiction shows that G = R = Zp
and F 6= cτ
ωformG = Np. Thus G satisfies Condition 1). Assume that
H(p) 6= ∅ and so 1 = G/CG(R) ∈ H(p). Hence G ∈ H. A contradiction.
Let G 6= CG(R), where R is an abelian p-group. Let’s consider
the group T = [R]M , where M = G/CG(R). Let C = CG(R). Then
C = C ∩ RM = R(C ∩ M). Evidently that (C ∩ M) is a normal sub-
group of G. But G is a monolithic group, then C ∩ M = 1 and so
R = CG(R) = Op(G) = Fp(G). It is not difficult to see that R = Cp(G)
and Op(G/Cp(G)) = Op(G/Op(G)) = Op(M) = 1 and so by Lemma 1.2,
Cp(T ) = R. Consequently by Lemma 1.3, sT ∈ F. Evidently that
|T | ≤ |G|. Now we suppose that |T | < |G|. Then T ∈ H and so
T/Cp(T ) ≃ T/R ≃ M ≃ G/CG(R) ≃ G/R ∈ H(p).
But G/Op(G) ⋍ G/R ∈ H and so G ∈ H by Lemma 1.3. This contra-
diction shows that T 6∈ H. Consequently T ∈ F\H. Thus in view of the
choice of G we have |T | = |G| and F = cτ
ωformT . It is clear that R = TH.
By Lemma 1.4,
f(p) = τform(T/Cp(T )) = τform(T/R) = τform(G/CG(R)) =
τform(G/R) = τformM.
Let M ∈ H(p). Consequently G ∈ NpH(p) = H(p). A contradiction.
Hence M /∈ H(p) and so τformM * H(p).
Let M be a proper τ -closed subformation of f(p). Assume that M *
H(p) and A be a group of minimal order in M\H(p). Since H(p) =
NpH(p), then Op(A) = 1. By Lemma 18.8 [5], exists a simple and faithful
Fp[A]-module P over Fp. Let F = [P ]A. Then P = CF (P ) = Op(F ) =
Cp(F ) and so
F/Op(F ) ≃ F/P ≃ A ∈ M ⊂ f(p) ⊆ f(p) ∩ F.
By Lemma 1.3, F ∈ F. Hence cτ
ωformF ⊆ F. If cτ
ωformF = F, then by
Lemma 1.4,
f(p) = τform(F/Cp(F )) = τform(F/P ) = τform(A) ⊆ M ⊂ f(p).
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.6 On minimal τ -closed ω-composition non-H-formations
This contradiction shows that cτ
ωformF ⊂ F. Then cτ
ωformF ⊆ H and
so F ∈ H. Hence F/Cp(F ) ≃ A ∈ H(p). A contradiction. Hence
M ⊆ H(p). Thus f(p) is a minimal τ -closed non-H(p)-formation.
Let M1 be a group of minimal order in τformM\H(p). Then M1 is
a monolithic τ -minimal non-H(p)-group with the socle Q = M
H(p)
1 and
τformM = τformM1.
Assume that Q ⊆ Φ(M1). Let t be the minimal 1-multiply local
satellite of H. By Theorem 8.3 [5], t is an inner satellite of H. Therefore
t(p) ⊆ H(p). Applying Theorem 8.3 [5] again and Consequence 8.6 [5] we
see that H(p) = Npt(p) is a local formation, as it is the product of two
local formations Np and t(p) (see Consequence 7.14 [5]). By Theorem 4.3
[11], H is a saturated formation. Since M1/Q ∈ H(p), then M1/Φ(M1) ∈
H(p). Consequently M1 ∈ H(p). This contradiction shows that Q *
Φ(M1).
Assume that p ∈ π(Com(Q)). Since M1/Q ∈ H(p), then M1 ∈
NpH(p) = H(p). This contradiction shows that Q is not a p-group.
Hence Op(M1) = 1. Thus there exists a simple and faithful Fp[M1]-
module R1 over Fp. Let G1 = [R1]M1. Hence R1 = CG1
(R1) = Op(G1) =
Cp(G1) = Fp(G1) is a minimal normal p-subgroup of G1 and so
G1/Op(G1) ≃ G1/R1 ≃ M1 ∈ τformM = f(p) ⊆ f(p) ∩ F.
By Lemma 1.3, G1 ∈ F.
Let H1 = cτ
ωformG1 and h1 be the minimal τ -valued ω-composition
satellite of H1. By Lemma 1.4,
h1(p) = τform(G1/Cp(G1)) = τform(G1/R1) = τform(M1).
If H1 ⊂ F, then H1 ⊆ H. Therefore by Lemma 1.5, h1 ≤ H, consequently,
M1 ≃ G1/R1 ∈ H(p).
This contradiction shows that H1 = F. Thus
F = cτ
ωformG1 = cτ
ωformG.
Now we shall show that G1 satisfies the hypothesis of the theorem.
In fact we have only to prove that R1 = GH
1 .
Indeed, if M1 ∈ H, then G1/R1 = G1/Cp(G1) ≃ M1 ∈ H(p). This
contradiction shows that G1 /∈ H. Consequently GH
1 = R1.
Let M1 /∈ H. Consequently cτ
ωformM1 = F. By Lemma 1.4,
f(p) = τform(M1/Cp(M1)) = τform(M1).
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.L. I. Belous, V. M. Sel’kin 7
But Q is not a p-group, so Q ⊆ Cp(M1). So τform(M1/Cp(M1)) ⊆
τform(M1/Q). Therefore τformM1 ⊆ τform(M1/Q). By Lemma 1.6,
M = τform(X ∪ {M1/Q}) is the unique maximal τ -closed subformation
of τformM1, where X is the set of all proper τ -subgroups of M1. Hence
M ⊂ τformM1. This contradiction shows that M1 /∈ H. Therefore
R1 = GH
1 .
Sufficiency. Let G be a group from the theorem. It is clear that
F * H.
Let π = ∅. In this case Oω(G) ∩ R(G) = 1. By Lemma 1.4,
f(ω′) = τform(G/(Oω ∩ R(G)) = τform(G).
Since G /∈ H, then
f(ω′) = τform(G) * H = H(ω′).
By Lemma 1.6, τform(X ∪ {G/R}) is the unique maximal τ -closed
subformation of f(ω′) = τformG, where X is the set of all proper τ -
subgroups of the group G. Since by hypothesis, all proper τ -subgroups
of G are contained in H, then
τform(X ∪ {G/R}) ⊆ H = H(ω′).
Hence all proper τ -closed subformations of f(ω′) are contained in H(ω′).
So f(ω′) is a minimal τ -closed non-H(ω′)-formation.
Let M be a proper τ -closed ω-composition subformation of F and m
be the minimal τ -valued ω-composition satellite of M. By Lemma 1.5,
m 6 f . We shall show that m 6 H.
Since
f(ω′) = τform(G) * m(ω′) = M,
consequently m(ω′) ⊂ f(ω′). Hence m(ω′) ⊆ H(ω′). Besides, since
G/R ∈ H, then G/R/Cq(G/R) ∈ H(q) for all q ∈ ω ∪ π(Com(G/R)).
By Lemma 1.1, Cq(G)/R = Cq(G/R) for all q ∈ ω. Consequently
G/Cq(G) ∈ H(q). Hence
m(q) ⊆ f(q) = τform(G/Cq(G)) ⊆ H(q).
Consequently m 6 H and so by Lemma 1.5, M ⊆ H. Thus F is a minimal
τ -closed ω-composition non-H-formation.
Let π 6= ∅ and p ∈ π.
If the group G satisfies Condition 1), then obviously, F is a minimal
τ -closed ω-composition non-H-formation.
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.8 On minimal τ -closed ω-composition non-H-formations
Let G satisfies Condition 2). By Lemma 1.4,
f(p) = τform(G/Cp(G)) = τform(G/R) = τform(M).
But M is a monolithic τ -minimal non-H(p)-group, then M /∈ H(p) and
so f(p) * H(p).
Let X be the set of all proper τ -subgroups of M . Therefore X ⊆ H(p).
But M/Q = M/MH(p) ∈ H(p). Hence
τform(X ∪ {M/Q}) ⊆ H(p).
By Lemma 1.6, τform(X∪{M/Q}) is the unique maximal τ -closed sub-
formation of f(p) = τform(M). Therefore all proper τ -closed subforma-
tions of f(p) are contained in H(p).
Consequently f(p) is a minimal τ -closed non-H(p)-formation, where
p ∈ π. We shall show that in this case the formation F is a minimal
τ -closed ω-composition non-H-formation.
Let M be a proper τ -closed ω-composition subformation of F and m
be the minimal τ -valued ω-composition satellite of M. By Lemma 1.5,
m 6 f . We shall show that m 6 H. Assume that m(p) = f(p). Then
G/Cp(G) = G/R = G/Op(G) ∈ m(p). Using now Lemma 1.3 we see
that G ∈ M and so
F = cτ
ωformG ⊆ M ⊂ F.
This contradiction shows that m(p) ⊂ f(p) and so from above we know
that m(p) ⊆ H(p). By Lemma 1.1, Cq(G)/R = Cq(G/R) for all prime
q 6= p and (R(G)∩Oω(G))/R = R(G/R)∩Oω(G/R). And since G/R ∈ H,
then f(ω′) ⊆ H(ω′) and f(q) ⊆ H(q) for all q ∈ ω\{p}. But m 6 f and
hence m(p) ⊆ H(p) for all p ∈ {ω′} ∪ ω. By Lemma 1.5, m 6 H.
Consequently M ⊆ H. Thus F is a minimal τ -closed ω-composition non-
H-formation.
Remark 1. If in Theorem 2.1 the formation H is those, that N ⊆ H,
then G cannot be a group of prime order.
Remark 2. If H is a τ -closed formation, then every minimal non-H-group
is a τ -minimal non-H-group.
Let’s note that in the case when τ is a trivial subgroup functor (i.e.
τ(G) = G for any group G) we obtain the following corollary:
Corollary 1. Let f be the minimal ω-composition satellite of the forma-
tion F and let H be the canonical ω-composition satellite of a 2-multiply
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.L. I. Belous, V. M. Sel’kin 9
local formation H. A formation F is a minimal ω-composition non-H-
formation if and only if F = cωformG, where G is a monolithic group
and R = GH = Soc(G) is the socle of G, where R * Φ(G) and either
π = π(Com(R)) ∩ ω = ∅ or π 6= ∅ and G satisfies one of the following
conditions:
1) G = R is a group of prime order p /∈ π(H);
2) G = [R]M , where R = Op(G) = Fp(G) for same p ∈ π and M is
a monolithic group and Q = MH(p) = Soc(M) is the socle of M , where
p 6∈ π(Com(Q)) and Q * Φ(M).
In the case when for all groups G the set τ(G) is the set of all sub-
normal subgroups of the group G instead of τ they write ssn.
Corollary 2. Let f be the minimal ssn-valued ω-composition satellite of
the formation F and let H be the canonical ω-composition satellite of a
2-multiply local formation H. A formation F is a minimal ssn-closed ω-
composition non-H-formation if and only if F = cssn
ω formG, where G is
a monolithic non-H-group and R = GH = Soc(G) is the socle of G, where
R * Φ(G) such that every popper subnormal subgroup of G belongs to H
and either π = π(Com(R)) ∩ ω = ∅ or π 6= ∅ and G satisfies one of the
following conditions:
1) G = R is a group of prime order p /∈ π(H);
2) G = [R]M , where R = Op(G) = Fp(G) for same p ∈ π and M is a
monolithic non-H(p)-group and Q = MH(p) = Soc(M) is the socle of M ,
where p 6∈ π(Com(Q)) and Q * Φ(M) such that every popper subnormal
subgroup of M belongs to H(p).
Corollary 3. Let S be the formation of all soluble groups. A formation
F is a minimal τ -closed ω-composition non-soluble formation if and only
if F = cτ
ωformG, where G is a monolithic τ -minimal non-soluble group
and R = GS = Soc(G) is the non-abelian socle of G.
Proof. Let H be the canonical ω-composition satellite of the formation
S. Hence H(a) = S for all a ∈ ω ∪ {ω′}.
Necessity. By Theorem 2.1 and Remark 1, F = cτ
ωformG, where G is
a monolithic τ -minimal non-S-group and R = GS * Φ(G) is the socle of
G and either π = π(Com(R)) ∩ ω = ∅ or π 6= ∅ and G = [R]M , where
R = Op(G) = Fp(G) for same p ∈ π and M is a monolithic τ -minimal
non-S-group and Q = MS is the socle of M , where Q * Φ(M).
Let’s π 6= ∅. In this case R is an abelian p-group. But G/R ∈ S
is a soluble group and so G is a soluble group. Then R = GS = 1. A
contradiction. Therefore R is a non-abelian group.
Sufficiency follows from Theorem 2.1.
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.10 On minimal τ -closed ω-composition non-H-formations
Corollary 4. Let N be the formation of all nilpotent groups. A formation
F is a minimal τ -closed ω-composition non-N-formation if and only if
F = cτ
ωformG, where G is a monolithic τ -minimal non-N-group and
R = GN = Soc(G) is the socle of G and either π = π(Com(R)) ∩ ω = ∅
or π 6= ∅ and G is a Schmidt group.
Proof. Let H be the canonical ω-composition satellite of the formation
H. Hence
H(a) =
{
Np, if a = p ∈ ω;
N, if a = ω′.
Necessity. By Theorem 2.1 and Remark 1, F = cτ
ωformG, where G is
a monolithic τ -minimal non-N-group and R = GN * Φ(G) is the socle of
G and either π = π(Com(R)) ∩ ω = ∅ or π 6= ∅ and G = [R]M , where
R = Op(G) = Fp(G) for same p ∈ π and M is a monolithic τ -minimal
non-H(p)-group and Q = MH(p) is the socle of M , where p 6∈ π(Com(Q))
and Q * Φ(M).
By Lemma 1.4,
f(p) = τform(G/Cp(G)) = τform(G/R) = τformM.
It means that τformM is a minimal τ -closed non-Np-formation. Since
G/R ≃ M ∈ N and N is hereditary, τformM ⊆ N. Thus by Theorem
2.4 [11], τformM = formM = sformM . Let H be a group of minimal
order in sformM\Np. If sformH ⊂ sformM , then sformH ⊆ Np.
A contradiction. Therefore sformH = sformM . By the choice of the
group H, it is a minimal non-Np-group. Thus all its Sylow subgroups are
p-groups. It means that H is p-group. A contradiction. Therefore H is
a group of prime order q, where q 6= p. Thus sformH = sformZq is
a hereditary formation generated by the group of prime order q. Since
M ∈ sformZq, M is a group of exponent q. Since G = [R]M and
R = CG(R), M is a irreducible abelian group of automorphisms for R.
Therefore M is a cyclic group. But the order and the exponent of the
cyclic group M are the same. Thus we have |M | = q. So G is a group
Schmidt.
Sufficiency. Let condition of the corollary be satisfied and R be an
abelian p-group. Hence G be a Schmidt group. From the description of
the Schmidt groups it follows that G = [R]M , where R = CG(R) is a
minimal normal p-subgroup of G and |M | = q, where q is a prime. It
means that M is a minimal non-Np-group and Q = M is the socle of M .
In this case Φ(M) = 1. Thus by Theorem 2.1, the corollary is proved.
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.L. I. Belous, V. M. Sel’kin 11
References
[1] Shemetov L.A., Screens of gratduated formations, Proc. VI All-Union Symposium
on the theory of groups, Kiev, Nauk. Dumka, 1980.
[2] Skiba A.N., Algebra of formations, Minsk, Belaruskaja Navuka, 1997.
[3] Skiba A.N., On critical formations, Vesti Akad. Navuk BSSR, N.4, 1980, pp.27-
33.
[4] Skiba A.N., On critical formations, In the book: "Infinite groups and related
algebraic structures", Kiev, 1993, pp.258-268
[5] Shemetkov L.A., Skiba A.N., Formation of algebraic systems. Moscow, Nauka,
1989.
[6] Rizhik V.N., On criticals of p-local formations, Problems in Algebra, 11, 1997,
pp.104-106.
[7] Safonova I.N., On minimal ω-local non-H-formations, Vesti Akad. Navuk Belarus.,
Ser. fiz.-mat. navuk, N.2, 1999, pp.23-27.
[8] Skiba A.N., Shemetkov L.A., Multiply L-composition formations of finite groups,
Ukrainsk. Math. Zh., 52(6), 2000, pp.783-797.
[9] Bliznets I.V., Critical ω-composition formation, Vesti Akad. Navuk Belarus., Ser.
fiz.-mat. navuk, N.4, 2002, pp.115-117.
[10] Doerk K., Hawkes T., Finite soluble grousp, Walter de gruyter, Berlin/New York,
1992.
[11] Shemetkov L.A., Formation of finite groups, Moscow, Nauka, 1978.
[12] Skiba A.N., Shemetkov L.A, On minimal composition screen of a composition
formation, Problems in Algebra, N.11, 1992, pp.39-43.
Contact information
L. I. Belous Gomel State University of F.Skorina, Be-
larus, 246019, Gomel, Sovetskaya Str., 104
E-Mail: LudaB@rambler.ru
V. M. Sel’kin Gomel State University of F.Skorina, Be-
larus, 246019, Gomel, Sovetskaya Str., 104
E-Mail: vsel’kin@gsu.unibel.by
Received by the editors: 06.05.2006
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