CLT-Groups with Hall S-Quasinormally Embedded Subgroups
A subgroup H of a finite group G is said to be Hall S-quasinormally embedded in G if H is a Hall subgroup of the S-quasinormal closure H SQG . We study finite groups G containing a Hall S-quasinormally embedded subgroup of index p n for each prime power divisor p n of the order of G.
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| author | Li, Shirong Liu, Jianjun Лі, Шіронг Лю, Жіанюнь |
| author_facet | Li, Shirong Liu, Jianjun Лі, Шіронг Лю, Жіанюнь |
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| description | A subgroup H of a finite group G is said to be Hall S-quasinormally embedded in G if H is a Hall subgroup of the S-quasinormal closure H SQG . We study finite groups G containing a Hall S-quasinormally embedded subgroup of index p n for each prime power divisor p n of the order of G. |
| first_indexed | 2026-03-24T02:20:41Z |
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| fulltext |
UDC 512.5
Jianjun Liu (School Math. and Statistics, Southwest Univ., China),
Shirong Li (Guangxi Univ., China)
CLT-GROUPS WITH HALL S-QUASINORMALLY EMBEDDED SUBGROUPS*
CLT-ГРУПИ З S-КВАЗIНОРМАЛЬНО ВКЛАДЕНИМИ ПIДГРУПАМИ ХОЛЛА
A subgroup H of a finite group G is said to be Hall S-quasinormally embedded in G if H is a Hall subgroup of the
S-quasinormal closure HSQG. We study finite groups G containing a Hall S-quasinormally embedded subgroup of index
pn for each prime power divisor pn of the order of G.
Пiдгрупа H скiнченної групи G називається пiдгрупою Холла, S-квазiнормально вкладеною в G, якщо H — пiдгрупа
Холла S-квазiнормального замикання HSQG. Вивчаються скiнченнi групи G, що мiстять S-квазiнормально вкладенi
пiдгрупи Холла iндексу pn для кожного простого степеневого дiльника pn порядку G.
1. Introduction. All groups considered in this paper are finite, our notation and terminology are
standard (see, for example, Robinson [25]).
A CLT-group is a group G of order n, say, having the property that for each divisor d of n,
there exists a subgroup in G of order d. Clearly, a CLT-group has Hall p′-subgroups for all primes
p, and hence it is solvable, but the converse is not true in general, the alternating group of degree 4
is an example of a non-CLT-group. Several years later, a nice extension was gave by T. M. Gagen
[13], which every solvable group can be embedded in a directly indecomposable CLT-group. Adding
requirements to the location or structure of the subgroup of order d yields various subclasses of
CLT-groups. In this aspect, C.V. Holmes first proved the following result.
Theorem 1.1 ([15], Theorem 1). A group G is nilpotent if and only if for each divisor d of the
order of G there exists a normal subgroup of order d.
Recently, S. R. Li, J. He, G. P. Nong and L. Q. Zhou [20] studied a new class of CLT-groups.
They introduced the following definition:
Definition 1.1 ([20], Definition 1). A subgroup H of a group G is called Hall normally embed-
ded in G if H is a Hall subgroup of the normal closure HG.
They studied the structure of a group G under the assumption that, for every factor d of the order
of G there exists a Hall normally embedded subgroup H of G of order d.
Some related topics can be found in [1, 3 – 11, 13 – 17, 19 – 24, 26 – 28, 30, 31] and [29] (Chapters
1, 4 and 6).
Recall that a subgroup H of a group G is S-quasinormal in G if HP = PH for all Sylow
subgroups P of G.
In this paper we analyze some results on the base of the following concept.
Definition 1.2. A subgroupH of a groupG is called a Hall S-quasinormally embedded subgroup
of G if H is a Hall subgroup of the S-quasinormal closure HSQG, the intersection of all the S-
quasinormal subgroups of G which contain H.
* The research of the work was supported by the National Natural Science Foundation of China (No. 11301426) and
(No. 11171243), the Natural Science Foundation Project of CQ (No. cstc2013jcyjA00019), the Fundamental Research
Funds for the Central Universities (No. XDJK2014B042).
c© JIANJUN LIU, SHIRONG LI, 2014
1146 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
CLT-GROUPS WITH HALL S-QUASINORMALLY EMBEDDED SUBGROUPS 1147
By definition, all S-quasinormal subgroups and all Hall subgroups (particularly Sylow subgroups)
of G are Hall S-quasinormally embedded in G. Clearly, a Hall normally embedded subgroup is
certainly a Hall S-quasinormally embedded subgroup, but the converse is not true in general, as the
following example shows:
Example 1.1. Let G = 〈a, b, c|a8 = b2 = c3 = 1, b−1ab = a−1, [a, c] = [b, c] = 1〉, then the
group G is a direct product of a dihedral group of order 16 and a cyclic group of order 3. We
write H = 〈b〉, generated by b. It is clear that H〈c〉 = 〈c〉H and so H is S-quasinormal in G. This
implies that H = HSQG and hence H is Hall S-quasinormally embedded in G. We can see that
HG = 〈a6, b〉. It means that H is not a Hall subgroup of HG. Thus H is not Hall normally embedded
in G.
In this paper, it is proved that, for each prime power divisor pn of the order of a group G, there
exists a Hall S-quasinormal embedded subgroup of index pn if and only if the nilpotent residual of
G is cyclic of square-free order.
2. Preliminaries. In this section we show some lemmas, which are required in Section 3.
Lemma 2.1 ([2], Theorems 1.2.14 and 1.2.19). Let G be a group. Then the following statements
hold:
(1) If K is a subgroup of G and H is S-quasinormal in G, then H ∩K is S-quasinormal in K.
(2) If H1 and H2 are two S-quasinormal subgroups of G, then H1 ∩H2 is S-quasinormal in G.
(3) If H is S-quasinormal in G, then H is subnormal in G.
Lemma 2.2. Let H be an S-quasinormal subgroup of a group G. Then Hg is also an S-
quasinormal subgroup of G, where g ∈ G.
Proof. This is obtained by direct checking.
Lemma 2.3. Let H be a subgroup of a group G. If there exists an S-quasinormal subgroup K
of G containing H such that H is a Hall subgroup of K, then H is Hall S-quasinormally embedded
in G.
Proof. According to our hypothesis and Lemma 2.1, we can see that HSQG is a subgroup of K.
Hence H is a Hall subgroup of HSQG, as desired.
Lemma 2.4. Let H be a Hall S-quasinormally embedded subgroup of a group G. Then the
following statements hold:
(a) If H ≤ K ≤ G, then H is Hall S-quasinormally embedded in K.
(b) If N �G, then HN/N is Hall S-quasinormally embedded in G/N.
(c) If N is S-quasinormal in G, then H ∩N is Hall S-quasinormally embedded in G.
However,
(d) If N �G, then HN may not be Hall S-quasinormally embedded in G.
(e) If N �G and N ≤ K, then K/N is Hall S-quasinormally embedded in G/N does not imply
that K is Hall S-quasinormally embedded in G.
Proof. (a) SinceH is a Hall subgroup ofHSQG, H is a Hall subgroup ofHSQG∩K. Furthermore,
HSQK ≤ HSQG ∩K and HSQG ∩K is S-quasinormal in K by Lemma 2.1. It follows from Lemma
2.3 that H is a Hall S-quasinormally embedded subgroup of K.
(b) Let π denote the set of prime factors of the order of H. Then H is a π-group and |HSQG :
H| is a π′-number. As (HN)SQG ≤ HSQGN, we can see that |(HN)SQG : HN | 6 |HSQGN :
HN | = |HSQG : H|
/
|HSQG∩N : H ∩N |, which is a π′-number. Hence HN/N is a Hall subgroup
of (HN/N)SQG and therefore HN/N is Hall S-quasinormally embedded in G/N.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
1148 JIANJUN LIU, SHIRONG LI
(c) It is clear that H ∩N ≤ (H ∩N)SQG ≤ HSQG ∩N and HSQG ∩N is S-quasinormal in G
by Lemma 2.1. On the other hand, H is a Hall subgroup of HSQG. It follows that H ∩N is a Hall
subgroup of HSQG ∩ N. Applying Lemma 2.3, we conclude that H ∩ N is Hall S-quasinormally
embedded in G.
(d) Let G = 〈a, b, c, d | a3 = b2 = c3 = d2 = 1, b−1ab = a−1, d−1cd = c−1, [a, c] = [a, d] =
= [b, c] = [b, d] = 1〉, then the group G is a direct product of two symmetric groups of degree 3.
We write H = 〈b〉, generated by b, then HSQG = 〈a, b〉. It is clear that H is a Hall subgroup of
HSQG. That is, H is Hall S-quasinormally embedded in G. Take N = 〈c, d〉 to be the subgroup of G
generated by c and d, then N is normal in G. We can see that HN = 〈b, c, d〉 and (HN)SQG = G,
which means that HN is not a Hall subgroup of (HN)SQG. Thus HN is not Hall S-quasinormally
embedded in G.
(e) Consider a group K = HN as in the proof of (d). We can identify that K/N is a Sylow 2-
subgroup of G/N and hence K/N is a Hall S-quasinormally embedded subgroup of G/N. However,
K is not Hall S-quasinormally embedded in G.
Lemma 2.4 is proved.
Lemma 2.5. LetH be an S-quasinormal subgroup of a solvable groupG. If p is a prime dividing
the order of G and H1 is a Hall p′-subgroup of G containing in H, then G = NG(H1)H.
Proof. Applying Lemma 2.1, H is subnormal in G. Therefore, there exists a subgroups series
H = G0 ≤ G1 ≤ . . . ≤ Gn = G
such that Gi � Gi+1, where 0 ≤ i ≤ n − 1. We prove the lemma by induction on n and suppose
that it has already been shown that Gi = NGi(H1)H for some i ∈ {1, 2, . . . , n− 1}. By the Frattini
argument, Gi+1 = NGi+1(H1)Gi = NGi+1(H1)H. This completes the induction argument.
Lemma 2.5 is proved.
Let N denote the class of all nilpotent groups, then N is a saturated formation. We denote by
GN the nilpotent residual of a group G.
Lemma 2.6. Let H be a subgroup of a group G. Then HN ≤ GN .
Proof. Since H
/
(H ∩GN ) ∼= HGN
/
GN ≤ G/GN is nilpotent, we can see that HN ≤ GN .
3. Main results. In this section, we study the structure of a group G when some subgroups are
Hall S-quasinormally embedded in G. Our first result is about supersolvability.
Theorem 3.1. For each prime power divisor pn of the order of a group G, if there exists a Hall
S-quasinormally embedded subgroup of G of index pn, then G is supersolvable.
Proof. The proof will follow as a consequence of the following steps.
1. Every Hall subgroup M of G satisfies the hypothesis of the theorem. Let π = π(M), the
set of primes of dividing |M |. Set pn
∣∣|M |. By hypothesis, there exists a subgroup H of G of index
pn such that H is Hall S-quasinormally embedded in G. Let H1 is a Hall π-subgroup of H. Then
from [25] (Theorem 9.1.7) it follows that Hg
1 ≤ M, for some g ∈ G. We can conclude that |M :
Hg
1 | = pn. To finish the proof of the statement it is enough to check that Hg
1 is Hall S-quasinormally
embedded in M. In fact, if H < HSQG, then from (|HSQG : H|, |H|) = 1 and |HSQG : H|
∣∣|G :
H| we obtain that H is a Hall subgroup of G. It follows that Hg
1 is a Hall subgroup of M, as
desired. If H = HSQG, then from Lemma 2.1 H is S-quasinormal in G and so is Hg by Lemma
2.2. Applying Lemma 2.1 again, Hg ∩M is S-quasinormal in M. Moreover, we can see that Hg
1 is
a Hall subgroup of Hg ∩M, hence Hg
1 is Hall S-quasinormally embedded in M by Lemma 2.3.
2. Let p be the smallest prime dividing the order of G, then G is p-nilpotent. Let P ∈ Sylp(G).
If p2†|G|, then by a theorem of Burnside [18] (IV, 2.8 Satz), G is p-nilpotent. Hence we can assume
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
CLT-GROUPS WITH HALL S-QUASINORMALLY EMBEDDED SUBGROUPS 1149
that P is not a cyclic group. By hypothesis, there exists a subgroup H of G of index |P |/p such
that H is a Hall subgroup of HSQG. Now the Burnside’s theorem [18] (IV, 2.8 Satz) implies that
H is p-nilpotent. If H = HSQG, then H is subnormal in G by Lemma 2.1, this means that G
is p-nilpotent. If H < HSQG, then it follows from (|HSQG : H|, |H|) = 1 and |HSQG : H|
∣∣|G :
H| that H is a Hall p′-subgroup, it is impossible.
3. G possesses Sylow tower of supersolvable type. Let K be a normal p-complement of G and
qm a prime power divisor of the order of K, where q 6= p. By hypothesis, there exists a subgroup H
of G of index qm such that H is Hall S-quasinormally embedded in G. It follows from (|G : K|, |G :
H|) = 1 that G = HK and hence |K : K ∩ H| = |G : H| = qm. By Lemma 2.4, K ∩ H is Hall
S-quasinormally embedded in K. So K satisfies the hypothesis. By induction, K possesses Sylow
tower of supersolvable type. Hence G possesses Sylow tower of supersolvable type, as desired.
4. Finish the proof. Let q be the largest prime divisor of the order ofG andQ ∈ Sylq(G). Then, by
hypothesis, G contains a subgroupH of index |Q|/q such thatH is a Hall subgroup ofHSQG.We can
argue as above to deduce that H = HSQG and hence H is S-quasinormal in G. Let H = H1(H∩Q),
where H1 is Hall q′-subgroup of H. In view of Lemma 2.5, G = NG(H1)H = NG(H1)(H ∩ Q).
We can see that HG
1 = H
NG(H1)(H∩Q)
1 = HH∩Q
1 ≤ H, it is clear that HG
1 = H or H1. If HG
1 = H,
then H �G. Since H1 is a Hall subgroup of G, then, by statement 1 and induction argument, H1 is
supersolvable and so is NG(H1). We can conclude that G
/
(H∩Q) is supersolvable and |H∩Q| = q,
this means thatG is supersolvable. If the latter is true, thenG = H1×Q and henceG is supersolvable.
Theorem 3.1 is proved.
We can now prove the following theorem.
Theorem 3.2. Let G be a group. Then the following statements are equivalent:
(a) For each prime power divisor pn of the order of G there exists a Hall S-quasinormally
embedded subgroup of G of index pn.
(b) For each divisor d of the order of G there exists a Hall S-quasinormally embedded subgroup
of G of order d.
(c) G = GNN with GN ∩N = 1, where GN is a cyclic group of square-free order.
(d) GN is cyclic of square-free order.
Proof. (a) ⇒ (b). Let |G|/d = pa11 p
a2
2 . . . pann , where ai > 0 and p1, p2, . . . , pn are distinct
primes. According to statement (a), there exists a Hall S-quasinormally embedded subgroup Bi of G
of index paii for all 1 ≤ i ≤ n. Set
H = B1 ∩B2 ∩ . . . ∩Bn.
Since all |G : Bi| are pairwise coprime, by [18] (I, 2.13 Hilssatz) we have
|G : H| = |G : B1||G : B2| . . . |G : Bn| = pa11 p
a2
2 . . . pann ,
whence |H| = d. To finishing our proof, we only need to show that H is a Hall subgroup of HSQG.
It is clear that Bi is a Hall subgroup of BSQG
i . If Bi < BSQG
i , then Bi is a Hall p′i-subgroup
of G. If Bi = BSQG
i , then Bi is S-quasinormal in G. Hence we may assume without loss of
generality that every element of {B1, B2, . . . , Bj} is a Hall subgroup of G and every element of
{Bj+1, Bj+2, . . . , Bn} is S-quasinormal in G, where 0 ≤ j ≤ n. Then
C = B1 ∩ . . . ∩Bj
is a Hall subgroup of G and
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
1150 JIANJUN LIU, SHIRONG LI
D = Bj+1 ∩ . . . ∩Bn
is a S-quasinormal subgroup of G. Moreover, H = C ∩D. Applying Lemma 2.4, we have H is Hall
S-quasinormally embedded in G.
(b) ⇒ (a). Clear.
(b) ⇒ (c). By Theorem 3.1, G is supersolvable. Let q be the largest prime dividing the order of
G and Q ∈ Sylq(G). Then, the theorem of Schur – Zassenhaus gives a complement K of Q in G and
G = KQ. The statement (c) will follow from the next four steps.
1. K satisfies the hypothesis of statement (b), by induction, K = KNK1 with KN ∩K1 = 1,
and the nilpotent residual KN is cyclic of square-free order. Let d be a divisor of the order of K.
By hypothesis, G contains a subgroup H of order d such that H is Hall S-quasinormally embedded
in G. In view of [25] (Theorem 9.1.7), Hg ≤ K for some g ∈ G. Notice that HSQG = L is
S-quasinormal in G, it follows from Lemma 2.2 that Lg is S-quasinormal in G and therefore Lg ∩K
is S-quasinormal in K by Lemma 2.1. Now the Lemma 2.3 may be applied to K to show that Hg is
Hall S-quasinormal embedded in K, as required.
2. G = K[K,Q] × CQ(K) = KQ1 × Q2, where Q1 = [K,Q], Q2 = CQ(K). In view of
[12] (Proposition 12.5), we can see that Q = [K,Q]CQ(K). By hypothesis, there exists a Hall
S-quasinormally embedded subgroup H of G of order q|K|. We can conclude that H = HSQG.
Write H = KQ1, where Q1 = H ∩ Q. Applying Lemma 2.5, G = NG(K)H = NG(K)Q1. If
Q1 ≤ NG(K), then K � G and therefore G = K × Q = K × CQ(K), as desired. Hence we
must only consider the case that Q1 � NG(K). Let Q2 = Q ∩ NG(K), then NG(K) = Q2 × K
and so G = KQ2Q1. In this case, [K,Q] = [K,Q2Q1] = [K,Q1] = Q1. Since Q2 is a maximal
subgroup of Q and K ≤ CG(Q2), both Q2 and KQ1 are normal in G. This implies that G =
= KQ1Q2 = K[K,Q]×Q2. Furthermore, we have CQ(K) = CG(K) ∩Q ≥ Q2. If CQ(K) > Q2,
then CQ(K) = Q and therefore G = K ×Q, in contradiction to the fact that Q1 � NG(K). Hence
CQ(K) = Q2, as desired.
3. GN = KNQ1. Obviously, K normalizes KN and Q1. It follows from G = KQ1Q2 that
KNQ1 is normal in G. We obtain that
G
/
KNQ1 = KQ1
/
KNQ1 ×KNQ1Q2
/
KNQ1
∼= K
/
KN ×KNQ1Q2
/
KNQ1
is nilpotent, which shows that GN ≤ KNQ1. Since |Q1| = q or 1, we see Q1 ∩GN = Q1 or 1. If
the latter is true, then, since G = G
/
(Q1 ∩GN ) . G
/
Q1×G
/
GN is q-nilpotent, we have that K is
normal in G and thus Q1 = [K,Q] = 1, Consequently, GN = KN by Lemma 2.6, as desired. Thus
we consider Q1 ≤ GN , in this case, KNQ1 ≤ GN and hence GN = KNQ1.
4. Finish the proof. By statement 2, KN is cyclic of square-free order and Q1 is of order
q or 1, it follows that GN is of square-free order. As G is supersolvable, we can see that G′ is
nilpotent by [18] (VI, 9.1 Satz). Moreover, GN ≤ G′ and so GN is cyclic. Let N = K1Q2, then
G = KQ1Q2 = GNK1Q2 = GNN, as desired.
(c)⇒ (d). Clear.
(d)⇒ (b). Let |G|/d = pa11 p
a2
2 . . . pann . Without generality, we may assume that every element of
{p1, p2, . . . , pi} is not a prime divisor of the order of GN , every element of {pi+1, pi+2, . . . , pn} is
a prime divisor of GN , where 0 ≤ i ≤ n. We can conclude that G is pk-nilpotent, where 1 ≤ k ≤ i.
Hence there exists a normal subgroup Hk of G such that |G : Hk| = pakk . Let pj
∣∣|GN |, where
i+1 ≤ j ≤ n. If p
aj+1
j † |G|, then, since G is solvable, it follows that G contains a Hall p′j-subgroup
Hj and therefore Hj is Hall S-quasinormally embedded in G. Now we may assume that p
aj+1
j
∣∣|G|.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
CLT-GROUPS WITH HALL S-QUASINORMALLY EMBEDDED SUBGROUPS 1151
As G/GN is nilpotent and GN is of square-free order, we deduce that there exists a normal subgroup
Hj/G
N of G/GN such that |G : Hj | = p
aj
j . Thus there exists a subgroup Hj of G of index p
aj
j such
that Hj is Hall S-quasinormally embedded in G in these two cases. Put
H = H1 ∩H2 ∩ . . . ∩Hn.
Since all |G : Hi| are pairwise coprime, by [18] (I, 2.13 Hilssatz) we have
|G : H| = |G : H1||G : H2| . . . |G : Hn| = pa11 p
a2
2 . . . pann .
It is clear that |H| = d. By Lemma 2.4, H is a Hall S-quasinormally embedded in G.
Theorem 3.2 is proved.
For convenience, we can give the following definition:
A group G is called an SEG-group if for each prime power divisor pn of the order of G, there
exists a Hall S-quasinormally embedded subgroup of G of index pn.
Notice that the class of CLT-groups is not closed under taking subgroups and quotient groups in
general. However, we have the following theorem.
Theorem 3.3. Let G be an SEG-group. Then the following statements are true:
(1) every subgroup of G is an SEG-group;
(2) every epimorphic image of G is an SEG-group.
Proof. (1) Let K ≤ G, then KN ≤ GN by Lemma 2.6. It follows from Theorem 3.2 that GN
is cyclic of square-free order and so is KN . Applying Theorem 3.2 again, K is an SEG-group.
(2) Let N be a normal subgroup of G. It follows from Theorem 3.2 and (G/N)N = GNN
/
N ∼=
∼= GN
/
GN ∩N that (G/N)N is cyclic of square-free order. Again by Theorem 3.2, we can see that
G/N is an SEG-group.
Theorem 3.3 is proved.
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after revision — 10.11.13
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| id | umjimathkievua-article-2205 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:20:41Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/6f/d699cf2e6ff7d8a0e1beda7b79b7116f.pdf |
| spelling | umjimathkievua-article-22052019-12-05T10:26:14Z CLT-Groups with Hall S-Quasinormally Embedded Subgroups CLT-групи з S-квазінормально вкладеними підгрупами Холла Li, Shirong Liu, Jianjun Лі, Шіронг Лю, Жіанюнь A subgroup H of a finite group G is said to be Hall S-quasinormally embedded in G if H is a Hall subgroup of the S-quasinormal closure H SQG . We study finite groups G containing a Hall S-quasinormally embedded subgroup of index p n for each prime power divisor p n of the order of G. Підгрупа $H$ скінченної групи $G$ називається підгрупою Холла, $S$-квазінормально вкладеною в $G$, якщо $H$ — підгрупа Холла $S$-квазінормального замикання $H^{SQG}$. Вивчаються скінченні групи $G$, що містять $S$-квазінормально вкладені підгрупи Холла індексу $p^n$ для кожного простого степеневого дільника $p^n$ порядку $G$. Institute of Mathematics, NAS of Ukraine 2014-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2205 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 8 (2014); 1146–1152 Український математичний журнал; Том 66 № 8 (2014); 1146–1152 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2205/1411 https://umj.imath.kiev.ua/index.php/umj/article/view/2205/1412 Copyright (c) 2014 Li Shirong; Liu Jianjun |
| spellingShingle | Li, Shirong Liu, Jianjun Лі, Шіронг Лю, Жіанюнь CLT-Groups with Hall S-Quasinormally Embedded Subgroups |
| title | CLT-Groups with Hall S-Quasinormally Embedded Subgroups |
| title_alt | CLT-групи з S-квазінормально вкладеними підгрупами Холла |
| title_full | CLT-Groups with Hall S-Quasinormally Embedded Subgroups |
| title_fullStr | CLT-Groups with Hall S-Quasinormally Embedded Subgroups |
| title_full_unstemmed | CLT-Groups with Hall S-Quasinormally Embedded Subgroups |
| title_short | CLT-Groups with Hall S-Quasinormally Embedded Subgroups |
| title_sort | clt-groups with hall s-quasinormally embedded subgroups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2205 |
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