A new characterization of finite σ-soluble PσT-groups
Let σ = {σi | i ∈ I} be a partition of the set of all primes ℙ and G a finite group. G is said to be σ-soluble if every chief factor H/K of G is a σᵢ-group for some i = i(H/K). A set H of subgroups of G is said to be a complete Hall σ-set of G if every member ≠ 1 of H is a Hall σᵢ-subgroup of G for...
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irk-123456789-1884992023-03-04T01:27:00Z A new characterization of finite σ-soluble PσT-groups Adarchenko, N.M. Let σ = {σi | i ∈ I} be a partition of the set of all primes ℙ and G a finite group. G is said to be σ-soluble if every chief factor H/K of G is a σᵢ-group for some i = i(H/K). A set H of subgroups of G is said to be a complete Hall σ-set of G if every member ≠ 1 of H is a Hall σᵢ-subgroup of G for some σᵢ∈ σ and H contains exactly one Hall σᵢ-subgroup of G for every i such that σᵢ ∩ π(G) ≠ ∅. A subgroup A of G is said to be σ-quasinormal or σ-permutable in G if G has a complete Hall σ-set H such that AHˣ = HˣA for all x ∈ G and all H ∈ H. We obtain a new characterization of finite σ-soluble groups G in which σ-permutability is a transitive relation in G. 2020 Article A new characterization of finite σ-soluble PσT-groups / N.M. Adarchenko // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 1. — С. 33–41. — Бібліогр.: 18 назв. — англ. 1726-3255 DOI:10.12958/adm1530 2010 MSC: 20D10, 20D15, 20D30 http://dspace.nbuv.gov.ua/handle/123456789/188499 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let σ = {σi | i ∈ I} be a partition of the set of all primes ℙ and G a finite group. G is said to be σ-soluble if every chief factor H/K of G is a σᵢ-group for some i = i(H/K). A set H of subgroups of G is said to be a complete Hall σ-set of G if every member ≠ 1 of H is a Hall σᵢ-subgroup of G for some σᵢ∈ σ and H contains exactly one Hall σᵢ-subgroup of G for every i such that σᵢ ∩ π(G) ≠ ∅. A subgroup A of G is said to be σ-quasinormal or σ-permutable in G if G has a complete Hall σ-set H such that AHˣ = HˣA for all x ∈ G and all H ∈ H. We obtain a new characterization of finite σ-soluble groups G in which σ-permutability is a transitive relation in G. |
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Adarchenko, N.M. A new characterization of finite σ-soluble PσT-groups Algebra and Discrete Mathematics |
| author_facet |
Adarchenko, N.M. |
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Adarchenko, N.M. |
| title |
A new characterization of finite σ-soluble PσT-groups |
| title_short |
A new characterization of finite σ-soluble PσT-groups |
| title_full |
A new characterization of finite σ-soluble PσT-groups |
| title_fullStr |
A new characterization of finite σ-soluble PσT-groups |
| title_full_unstemmed |
A new characterization of finite σ-soluble PσT-groups |
| title_sort |
new characterization of finite σ-soluble pσt-groups |
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Інститут прикладної математики і механіки НАН України |
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2020 |
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http://dspace.nbuv.gov.ua/handle/123456789/188499 |
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A new characterization of finite σ-soluble PσT-groups / N.M. Adarchenko // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 1. — С. 33–41. — Бібліогр.: 18 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT adarchenkonm anewcharacterizationoffinitessolublepstgroups AT adarchenkonm newcharacterizationoffinitessolublepstgroups |
| first_indexed |
2025-07-16T10:35:21Z |
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2025-07-16T10:35:21Z |
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1837799437287555072 |
| fulltext |
“adm-n1” — 2020/5/14 — 19:35 — page 33 — #41
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 29 (2020). Number 1, pp. 33–41
DOI:10.12958/adm1530
A new characterization of finite σ-soluble
PσT -groups
N. M. Adarchenko
Communicated by L. A. Kurdachenko
Abstract. Let σ = {σi | i ∈ I} be a partition of the set
of all primes P and G a finite group. G is said to be σ-soluble if
every chief factor H/K of G is a σi-group for some i = i(H/K).
A set H of subgroups of G is said to be a complete Hall σ-set
of G if every member 6= 1 of H is a Hall σi-subgroup of G for
some σi ∈ σ and H contains exactly one Hall σi-subgroup of G for
every i such that σi ∩ π(G) 6= ∅. A subgroup A of G is said to
be σ-quasinormal or σ-permutable in G if G has a complete Hall
σ-set H such that AHx = HxA for all x ∈ G and all H ∈ H. We
obtain a new characterization of finite σ-soluble groups G in which
σ-permutability is a transitive relation in G.
1. Introduction
Throughout this paper, all groups are finite and G always denotes
a finite group. Moreover, P is the set of all primes, π ⊆ P and π′ = P \ π.
The group G is called π-supersoluble provided every chief factor of G is
either cyclic or a π′-group. If n is an integer, the symbol π(n) denotes the
set of all primes dividing n; as usual, π(G) = π(|G|), the set of all primes
dividing the order of G.
2010 MSC: 20D10, 20D15, 20D30.
Key words and phrases: finite group, σ-permutable subgroup, PσT -group,
σ-soluble group, σ-nilpotent group.
https://doi.org/10.12958/adm1530
“adm-n1” — 2020/5/14 — 19:35 — page 34 — #42
34 New characterization of finite σ-soluble PσT -groups
In what follows, σ is some partition of P, that is, σ = {σi|i ∈ I}, where
P =
⋃
i∈I σi and σi ∩ σj = ∅ for all i 6= j. The symbol σ(n) denotes the
set {σi|σi ∩ π(n) 6= ∅}; σ(G) = σ(|G|).
The group G is said to be: σ-primary (A.N. Skiba [1]) if G is a σi-group
for some i ∈ I; σ-decomposable (L.A. Shemetkov [2]) or σ-nilpotent [3, 4]
if G = G1 × · · · ×Gn for some σ-primary groups G1, . . . , Gn; σ-soluble [1]
if every chief factor of G is σ-primary.
A set H of subgroups of G is a complete Hall σ-set of G [3, 5] if every
member 6= 1 of H is a Hall σi-subgroup of G for some σi ∈ σ and H
contains exactly one Hall σi-subgroup of G for every σi ∈ σ(G).
Let τH(A) = {σi ∈ σ(G) \ σ(A) | σ(A) ∩ σ(HG) 6= ∅ for a Hall
σi-subgroup H ∈ H}.
Then we say, following Beidleman and Skiba [6], that a subgroup A
of G is: (i) τσ-permutable in G with respect to H if AHx = HxA for all
x ∈ G and all H ∈ H such that σ(H) ⊆ τH(A); (ii) τσ-permutable in G if
A is τσ-permutable in G with respect to some complete Hall σ-set H of G.
Recall also that a subgroup A of G is said to be: σ-permutable in G [1]
if G possesses a complete Hall σ-set H such that AHx = HxA for all
H ∈ H and all x ∈ G; σ-semipermutable in G [7] if G possesses a complete
Hall σ-set H such that AHx = HxA for all x ∈ G and all H ∈ H with
σ(A) ∩ σ(H) = ∅; σ-subnormal in G [1] if there is a subgroup chain
A = A0 6 A1 6 · · · 6 At = G
such that either Ai−1 E Ai or Ai/(Ai−1)Ai
is σ-primary for all i = 1, . . . , t.
In the classical case when σ = σ1 = {{2}, {3}, . . .} (we use here the
notations in [3]), σ-permutable, σ-semipermutable and τσ-quasinormal
subgroups are also called respectively S-permutable [8], S-semipermutable
[9] and τ -permutable [10,11], and in this case σ-subnormal subgroups are
exactly subnormal subgroups of the group.
It is clear that every σ-permutable subgroup is also σ-semipermutable
and every σ-semipermutable subgroup is τσ-permutable.
Recall also that G is said to be a PσT -group [1] if σ-permutability is
a transitive relation in G, that is, if H is a σ-permutable subgroup of K
and K is a σ-permutable subgroup of G, then H is σ-permutable in G.
In the case when σ = σ1, a PσT -group is called a PST -group [8].
In view of Theorem B in [1], PσT -groups can be characterized as the
groups in which every σ-subnormal subgroup is σ-permutable. Another
characterizations of PσT -groups are obtained in the papers [3, 7, 12–14].
“adm-n1” — 2020/5/14 — 19:35 — page 35 — #43
N. M. Adarchenko 35
Our main goal here is to give a characterization of PσT -groups in the
terms of τσ-permutable subgroups.
Theorem 1.1. Let D = GNσ and π = π(D). Suppose that G possesses
a complete σ-set H all members of which are π-supersoluble. Then G is a σ-
soluble PσT -group if and only if every σi-subgroup of G is τσ-permutable
in G for all σi ∈ σ(D).
In this theorem the symbol GNσ denotes the σ-nilpotent residual of G,
that is, the intersection of all normal subgroups N of G with σ-nilpotent
quotient G/N ; GN is the nilpotent residual of G.
Corollary 1.1 (see Theorem A in [7]). Let D = GNσ and π = π(D).
Suppose G possesses a complete Hall σ-set H all members of which are
π-supersoluble. If every σi-subgroup of G is σ-semipermutable in G for all
σi ∈ σ(D), then G is a σ-soluble PσT -group.
As another application of Theorem 1.1, we give the following new
characterization of soluble PST -groups.
Corollary 1.2. G is a soluble PST -group if and only if every subgroup of
every Sylow p-subgroup of G is τ -semipermutable in G for all p ∈ π(GN).
All unexplained notation and terminology are standard. The reader is
referred to [8], [15] or [16] if necessary.
2. Preliminaries
We use Nσ to denote the class of all σ-nilpotent groups.
Lemma 2.1 (see Lemma 2.5 in [1]). The class Nσ is closed under taking
direct products, homomorphic images and subgroups. Moreover, if E is
a normal subgroup of G and E/E ∩ Φ(G) is σ-nilpotent, then E is σ-
nilpotent.
In view of Proposition 2.2.8 in [16], we get from Lemma 2.1 the
following
Lemma 2.2. If N is a normal subgroup of G, then
(G/N)Nσ = GNσN/N.
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36 New characterization of finite σ-soluble PσT -groups
Lemma 2.3 (V. N. Knyagina and V. S. Monakhov [17]). Let H, K and
N be pairwise permutable subgroups of G and H a Hall subgroup of G.
Then
N ∩HK = (N ∩H)(N ∩K).
Recall that G is said to be: a Dπ-group if G possesses a Hall π-subgroup
E and every π-subgroup of G is contained in some conjugate of E; a σ-full
group of Sylow type [1] if every subgroup E of G is a Dσi
-group for every
σi ∈ σ(E); σ-full [5] provided G possesses a complete Hall σ-set.
In view of Theorems A and B in [5], the following fact is true.
Lemma 2.4. If G is σ-soluble, then G is a σ-full group of Sylow type.
Lemma 2.5 (see Lemma 3.1 in [1]). Let H be a σi-subgroup of a σ-full
group G. Then H is σ-permutable in G if and only if Oσi(G) 6 NG(H).
Lemma 2.6. Suppose that G is σ-full and D := GNσ is a nilpotent Hall
subgroup of G. If every σi-subgroup of G is τσ-permutable in G for all
σi ∈ σ(D), then every subgroup of D is normal in G.
Proof. Suppose that this lemma is false and let G be a counterexam-
ple of minimal order. By hypothesis, G possesses a complete Hall σ-set
{H1, . . . , Ht}. We can assume without loss of generality that Hi is a σi-
group for all i = 1, . . . , t.
First we show that the hypothesis holds on G/N for every minimal
normal subgroup N of G. First note that
(G/N)Nσ = DN/N ≃ D/(D ∩N)
is a nilpotent Hall subgroup of G/N by Lemma 2.2.
Now let V/N be a non-identity σi-subgroup of G/N for some
σi ∈ σ((G/N)Nσ) = σ(DN/N) = σ(D/(D ∩N)) ⊆ σ(D).
And let U be a minimal supplement to N in V . Then U ∩N 6 Φ(U), so
U is a σi-subgroup of G since V/N = UN/N ≃ U/(U ∩ N). Therefore
U is τσ-permutable in G by hypothesis and σ(U) = σ(UN/N) = {σi},
which implies that V/N = UN/N is τσ-quasinormal in G/N by Lemma
2.6(1) in [6]. Hence the hypothesis holds on G/N .
Now let H be a subgroup of the Sylow p-subgroup P of D for some
prime p ∈ π. We show that H is normal in G. For some i we have
P 6 Oσi
(D) = Hi∩D. On the other hand, we have D = Oσi
(D)×Oσi(D)
“adm-n1” — 2020/5/14 — 19:35 — page 37 — #45
N. M. Adarchenko 37
since D is nilpotent. Assume that Oσi(D) 6= 1 and let N be a minimal
normal subgroup of G contained in Oσi(D). Then HN/N 6 DN/N =
(G/N)Nσ , so the choice of G implies that HN/N is normal in G/N . Hence
H = H(N ∩Oσi
(D)) = HN ∩Oσi
(D) is normal in G.
Now assume that Oσi(D) = 1, so D is a σi-group. Since G/D is
σ-nilpotent by Lemma 2.1, Hi/D is normal in G/D and hence Hi is
normal in G. Therefore all subgroups of Hi are σ-permutable in G by
Lemma 2.6(3) in [6] and hypothesis. Since D is a normal Hall subgroup
of Hi, it has a complement S in Hi by the Schur-Zassenhaus theorem.
Lemma 2.5 implies that D 6 Oσi(G) 6 NG(S). Hence Hi = D×S. Hence
S 6 NG(H), so
G = HiO
σi(G) = (SD)Oσi(G) = SOσi(G) 6 NG(H),
so H is normal in G. Therefore every subgroup of D is normal in G since
D is nilpotent by hypothesis. The lemma is proved.
Lemma 2.7 (see Theorem A in [3]). Let D = GNσ . If G is a σ-soluble
PσT -group, then the following conditions hold:
(i) G = D ⋊ M , where D is an abelian Hall subgroup of G of odd
order, M is σ-nilpotent and every element of G induces a power
automorphism in D;
(ii) Oσi
(D) has a normal complement in a Hall σi-subgroup of G for
all i.
Conversely, if Conditions (i) and (ii) hold for some subgroups D and
M of G, then G is a PσT -group.
Lemma 2.8 (see Theorem A in [3]). The following statements hold:
(1) G is a PσT -group if and only if every σ-subnormal subgroup of G
is σ-permutable in G;
(2) If G is a σ-soluble PσT -group, then every quotient G/N of G is
also a σ-soluble PσT -group.
3. Proof of Theorem 1.1
Let H = {H1, . . . , Ht}. We can assume without loss of generality that
Hi is a σi-group for all i = 1, . . . , t.
First show that if every σi-subgroup of G is τσ-permutable in G for
all σi ∈ σ(D), then G is a σ-soluble PσT -group. Assume that this is false
and let G be a counterexample of minimal order.
“adm-n1” — 2020/5/14 — 19:35 — page 38 — #46
38 New characterization of finite σ-soluble PσT -groups
(1) G = D⋊M , where D is an abelian Hall subgroup of G of odd order,
M is σ-nilpotent and every element of G induces a power automorphism in
D (this claim directly follows from Lemma 2.6 and Theorem 1.5 in [18]).
(2) If R is a non-identity normal subgroup of G, then the hypothesis
holds for G/R, so G/R is a σ-soluble PσT -group.
First note that
H0 = {H1N/N, . . . ,HtN/N}
is a complete Hall σ-set of G/N . Moreover, every member HiN/N ≃
Hi/(Hi∩N) of H0 is π-supersoluble since Hi is π-supersoluble by hypoth-
esis. On the other hand, (G/N)Nσ = DN/N ≃ D/(D ∩ N) by Lemma
2.2. Hence π0 ⊆ π, where π0 = π((G/N)Nσ), so every member of H0 is
π0-supersoluble.
Let V/N be a non-identity σi-subgroup of G/N for some
σi ∈ σ((G/N)Nσ) = σ(DN/N) = σ(D/D ∩N) ⊆ σ(D).
And let U be a minimal supplement to N in V . Then U ∩N 6 Φ(U), so
U is a σi-subgroup of G by the isomorphism V/N = UN/N ≃ U/U ∩N .
Therefore U is τσ-permutable in G by hypothesis and σ(U) = σ(UN/N) =
{σi}, which implies that V/N = UN/N is τσ-permutable in G/N by
Lemma 2.6(1) in [6]. Hence the hypothesis holds on G/N , so the choice
of G implies that G/N is a σ-soluble PσT -group.
(3) Hi = Oσi
(D)× S for some subgroup S of Hi for each σi ∈ σ(D).
Since D is a nilpotent Hall subgroup of G by Claim (1), D = L×N ,
where L = Oσi
(D) and N = Oσi(D) are Hall subgroups of G. First assume
that N 6= 1. Then
Oσi
((G/N)Nσ) = Oσi
(D/N) = LN/N
has a normal complement V/N in HiN/N ≃ Hi by Claim (2). On the
other hand, N has a complement S in V by the Schur-Zassenhaus theorem.
Hence Hi = Hi ∩ LSN = LS and L ∩ S = 1 since
(L ∩ S)N/N 6 (LN/N) ∩ (V/N) = (LN/N) ∩ (SN/N) = 1.
It is clear that V/N is a Hall subgroup of HiN/N , so V/N is characteristic
in HiN/N . On the other hand, HiN/N is normal in G/N by Lemma 2.2
“adm-n1” — 2020/5/14 — 19:35 — page 39 — #47
N. M. Adarchenko 39
since D/N 6 HiN/N . Hence V/N is normal in G/N . Thus Hi ∩ V =
Hi ∩NS = S(Hi ∩N) = S is normal in Hi, so Hi = Oσi
(D)× S.
Now assume that D = Oσi
(D). Then Hi is normal in G, so all sub-
groups of Hi are σ-permutable in G by Lemma 2.6(3) in [6]. Since D is
a normal Hall subgroup of Hi, it has a complement S in Hi. Lemma 2.5
implies that D 6 Oσi(G) 6 NG(S). Hence Hi = D × S = Oσi
(D)× S.
Now, from Lemma 2.7 and Claims (2) and (3) it follows that G is
a σ-soluble PσT -group, contrary our assumption on the G. This completes
the proof of the sufficiency of the condition of the theorem.
Now we show that if G is a σ-soluble PσT -group, then every σi-
subgroup of G is τσ-permutable in G for each σi ∈ σ(D). It is enough
to show that H is a σi-subgroup of G, then H permutes with every Hall
σj-subgroups of G for all j 6= i.
Assume that this is false and let G be a counterexample of minimal
order. Then D 6= 1 and there are σi and σj (i 6= j) such that σi ∈ σ(D)
and HE 6= EH for some σi-subgroup H and some Hall σj-subgroup E
of G. Then H is not σ-subnormal in G by Lemma 2.8. Hence a Hall
σi-subgroup Hi of G is not normal in G since otherwise we have H 6 Hi
and so H is σ-subnormal in G by Lemma 2.6(6) in [1]. Now note that
|σ(D)| > 1, Indeed, if |σ(D)| = 1, then σ(D) = {σi} and so D 6 Hi,
which implies that Hi/D is normal in G/D since G/D is σ-nilpotent by
Lemma 2.1. But then Hi is normal in G, a contradiction.
Now we show that EHN is a subgroup of G for every minimal normal
subgroup N of G. First note that the hypothesis holds for G/N by Lemma
2.8. Moreover, HN/N ≃ H/H ∩N is a σi-subgroup of G/N . Therefore,
if σi ∈ σ(DN/N) = σ((G/N)Nσ), then the choice of G implies that
(HN/N)(EN/N) = (EN/N)(HN/N) = EHN/N
is a subgroup of G/N . Hence EHN is a subgroup of G. Now assume that
σi 6∈ σ(DN/N). Then a Hall σi-subgroup Hi of G is contained in N , so
Hi = N since N is σ-primary. But then H 6 N and so H is σ-subnormal
in G, a contradiction. Hence EHN is a subgroup of G.
Since |σ(D)| > 1 and D is abelian by Lemma 2.7, G has at least
two σ-primary minimal normal subgroups R and N such that R,N 6 D
and σ(R) 6= σ(N). Then at least one of the subgroups R or N , R say, is
a σk-group for some k 6= j. Moreover,
R ∩ E(HN) = (R ∩ E)(R ∩HN) = R ∩HN
“adm-n1” — 2020/5/14 — 19:35 — page 40 — #48
40 New characterization of finite σ-soluble PσT -groups
by Lemma 2.3 and R ∩ HN 6 Oσk
(HN) 6 V , where V is a Hall σk-
subgroup of H, since N is a σ′
k-group and G is a σ-full group of Sylow
type by Lemma 2.1. Hence
EHR ∩ EHN = E(HR ∩ EHN) = EH(R ∩ E(HN))
= EH(R ∩HN) = EH(R ∩H) = EH
is a subgroup of G. Hence HE = EH. This contradicts the fact that
HE 6= EH. The necessity of the condition of the theorem is proved.
The theorem is proved.
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[18] N.M. Adarchenko, On τσ-permutable subgroups of finite groups, Preprint, 2019.
Contact information
Nikita
M. Adarchenko
Department of Mathematics and Technologies
of Programming, Francisk Skorina Gomel State
University, Gomel 246019, Belarus
E-Mail(s): nik.adarchenko@gmail.com
Received by the editors: 20.01.2020.
N. M. Adarchenko
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