A note on $SΦ$-supplemented subgroups
We give new and brief proofs of the results obtained by X. Li and T. Zhao in [\mathrm{S}\Phi -supplemented subgroups of finite groups // Ukr. Math. J. – 2012. – 64, № 1. – P. 102–109].
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| author | Li, C. Yi, X. Zhang, Xirong Лі, К. Ій, Х. Чжан, Хіронг |
| author_facet | Li, C. Yi, X. Zhang, Xirong Лі, К. Ій, Х. Чжан, Хіронг |
| author_sort | Li, C. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:31:14Z |
| description | We give new and brief proofs of the results obtained by X. Li and T. Zhao in [\mathrm{S}\Phi -supplemented subgroups of finite groups
// Ukr. Math. J. – 2012. – 64, № 1. – P. 102–109]. |
| first_indexed | 2026-03-24T02:15:01Z |
| format | Article |
| fulltext |
UDC 512.5
C. Li (School Math. and Statistics, Jiangsu Normal Univ., Xuzhou, China),
X. Zhang (Yancheng Inst. Technology, China),
X. Yi (School Sci., Zhejiang Sci.-Techn. Univ., Hangzhou, China)
A NOTE ON \bfS \bfPhi -SUPPLEMENTED SUBGROUPS*
ПРО \bfS \bfPhi -ДОПОВНЕНI ПIДГРУПИ
We give new and brief proofs of the results obtained by X. Li and T. Zhao in [\mathrm{S}\Phi -supplemented subgroups of finite groups
// Ukr. Math. J. – 2012. – 64, № 1. – P. 102–109].
Наведено новi та короткi доведення результатiв, що були отриманi Х. Лi та Т. Жао в [\mathrm{S}\Phi -supplemented subgroups of
finite groups // Ukr. Math. J. – 2012. – 64, № 1. – P. 102 – 109].
All groups considered in this note will be finite. In [2], X. Li and T. Zhao called that a subgroup
H of a group G is \mathrm{S}\Phi -supplemented in G if there exists a subnormal subgroup T of G such that
G = HT and H \cap T \leq \Phi (H), where \Phi (H) is the Frattini subgroup of H . They investigated
the influence of \mathrm{S}\Phi -supplemented subgroups on the p-nilpotency, supersolvability and formation.
However, some conditions of many theorems are unnecessary and may been removed. Moreover, the
proofs are complicated. In the note, we prove some new results which can deduce their theorems and
consequently simplify their proofs.
1. Preliminaries.
Lemma 1.1. Let p be a fixed prime dividing the order of a group G and P a Sylow p-subgroup
of G. Suppose that all maximal subgroups of P are \mathrm{S}\Phi -supplemented in G. Then, either G is a
group whose Sylow p-subgroups are of order p, or G is a p-nilpotent group.
Proof. Suppose that | P | > p. Let M be a maximal subgroup of P . By the hypothesis, M
is \mathrm{S}\Phi -supplemented in G. Then there exists a subnormal subgroup T of G such that MT = G
and M \cap T \leq \Phi (M). Obviously, T < G. Since | G : T | is a power of p and T \lhd \lhd G, we have
Op(G) \leq T . Consequently, Op(G) < G and so Op(G) \cap P < P . We may choose a maximal
subgroup P1 of P such that Op(G) \cap P \leq P1 < P . Then Op(G) \cap P = Op(G) \cap P1 . Since P1 is
\mathrm{S}\Phi -supplemented in G, there is also a subnormal K such that Op(G) \leq K and P1 \cap K \leq \Phi (P1) as
above arguments. It follows that Op(G)\cap P = Op(G)\cap P1 \leq K \cap P1 \leq \Phi (P1) \leq \Phi (P ). By Tate’s
theorem (see [1], IV, 4.7), Op(G) is p-nilpotent. It follows that G is p-nilpotent.
Lemma 1.2. Let p be a fixed prime dividing the order of a group G and P a Sylow p-subgroup
of G. Suppose that P has a subgroup D such that 1 < | D| < | P | and every subgroup H of P with
| H| = | D| is \mathrm{S}\Phi -supplemented in G. Then G is p-nilpotent.
Proof. Suppose that | P : D| = p. Then, since p \leq | D| , we have that p2 \leq | P | . By Lemma 1.1,
the group G is p-nilpotent. Hence we may assume that | P : D| > p. Let H \leq P such that | H| = | D| .
By the hypothesis, H is \mathrm{S}\Phi -supplemented in G. Then there exists a subnormal subgroup T of G
such that HT = G and H \cap T \leq \Phi (H). Obviously, T < G. Hence G has a proper normal K such
that T \leq K . Since G/K is a p-group, G has a normal maximal subgroup M such that HM = G and
| G : M | = p. It is easy to see that M satisfies the hypotheses of the theorem. By induction, M is
p-nilpotent, and so G is p-nilpotent.
* This paper was supported by the Natural Science Foundation of China (No. 11401264 and 11571145) and the Priority
Academic Program Development of Jiangsu Higher Education Institutions.
c\bigcirc C. LI, X. ZHANG, X. YI, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8 1139
1140 C. LI, X. ZHANG, X. YI
Lemma 1.3. Let E be a normal subgroup of a group G. Suppose that for every prime p dividing
| E| , there exists a Sylow p-subgroup P of E such that P has a subgroup D satisfying 1 < | D| < | P |
and every subgroup H of P with | H| = | D| is \mathrm{S}\Phi -supplemented in G. Then every G-chief factor
of E is cyclic.
Proof. By [2] (Lemma 2.1(1)), every subgroup H of P with | H| = | D| is \mathrm{S}\Phi -supplemented
in E . Applying Lemma 1.2, E is p-nilpotent. Let Ep\prime be the normal p-complement of E .
Obviously, Ep\prime is normal in G. First, assume that Ep\prime \not = 1. It is easy to see that the hypotheses
of the theorem hold for (G/Ep\prime , E/Ep\prime ) and (G,Ep\prime ). By induction, E/Ep\prime \leq Z\scrU (G/Ep\prime ) and
Ep\prime \leq Z\scrU (G). Hence, E \leq Z\scrU (G). Now, assume that P = E . In view of [2] (Lemma 2.4), we may
assume | P : D| > p. Let H \leq P such that | H| = | D| . By the hypothesis, H is \mathrm{S}\Phi -supplemented
in G. Then there exists a subnormal subgroup T of G such that HT = G and H \cap T \leq \Phi (H). It
is easy to see that G has a normal maximal subgroup M such that HM = G and | G : M | = p. It
is easy to see that P \cap M satisfies the hypotheses of the theorem. By induction, P \cap M \leq Z\scrU (G).
Note that | P/P \cap M | = p, we have P \leq Z\scrU (G).
2. Brief proofs of results in [2].
Theorem 2.1 ([2], Theorem 3.1). Let P be a Sylow p-subgroup of a group G, where p is a
prime divisor of | G| such that (| G| , p - 1) = 1. If every maximal subgroup of P is \mathrm{S}\Phi -supplemented
in G, then G is p-nilpotent.
Proof. By Lemma 1.1, | P | = p or G is p-nilpotent. If | P | = p, then G is also p-nilpotent by
[4] (Lemma 2.6).
From Lemma 1.2, we arrive at the following Theorems 2.2 and 2.3.
Theorem 2.2 ([2], Theorem 3.2). Let G be a group and P a Sylow p-subgroup of G, where p is
a prime dividing | G| such that (| G| , p - 1) = 1. Let D be a subgroup of P such that 1 < | D| < | P | .
If every subgroup H of P with | H| = | D| is \mathrm{S}\Phi -supplemented in G, then G is p-nilpotent.
Theorem 2.3 ([2], Theorem 3.3). Let p be a prime dividing | G| and P a Sylow p-subgroup of
G. If NG(P ) is p-nilpotent and there exists a subgroup D of P such that 1 < | D| < | P | and every
subgroup H of P with | H| = | D| is \mathrm{S}\Phi -supplemented in G, then G is p-nilpotent.
Remark. From Lemma 1.2, the conditions that (| G| , p - 1) = 1 in Theorem 2.2 and NG(P ) is
p-nilpotent in Theorem 2.3 are unnecessary.
Theorem 2.4 ([2], Theorem 3.4). Let \scrF be a saturated formation containing \scrU and E a normal
subgroup of a group G such that G/E \in \scrF . If for every prime p dividing | E| , there exists a Sylow
p-subgroup P of E such that P has a subgroup D satisfying 1 < | D| < | P | and every subgroup H
of P with | H| = | D| is \mathrm{S}\Phi -supplemented in G, then G \in \scrF .
Proof. Since E \leq Z\scrU (G) by Lemma 1.3 and Z\scrU (G) \leq Z\scrF (G), we have E \leq Z\scrF (G) and so
G/Z\scrF (G) \sim = (G/E)/(Z\scrF (G)/E) \in \scrF . It follows that G \in \scrF .
Theorem 2.5 ([2], Theorem 3.5). Let \scrF be a saturated formation containing \scrU and E a normal
subgroup of a group G such that G/E \in \scrF . If for every prime p dividing | F \ast (E)| , there exists a
Sylow p-subgroup P of F \ast (E) such that P has a subgroup D satisfying 1 < | D| < | P | and every
subgroup H of P with | H| = | D| is \mathrm{S}\Phi -supplemented in G, then G \in \scrF .
Proof. Since F \ast (E) \leq Z\scrU (G) by Lemma 1.3 and Z\scrU (G) \leq Z\scrF (G), we have F \ast (E) \leq Z\scrF (G).
By [3] (Theorem B), E \leq Z\scrF (G). Since G/Z\scrF (G) \sim = (G/E)/(Z\scrF (G)/E) \in \scrF , we have G \in \scrF .
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
A NOTE ON \mathrm{S}\Phi -SUPPLEMENTED SUBGROUPS 1141
References
1. Huppert B. Endiche Gruppen I. – Berlin: Springer-Verlag, 1968.
2. Li X., Zhao T. \mathrm{S}\Phi -supplemented subgroups of finite groups // Ukr. Math. J. – 2012. – 64, № 1. – P. 102 – 109.
3. Skiba A. N. On two questions of L. A. Shemetkov concerning hypercyclically embedded subgroups of finite groups
// J. Group Theory. – 2010. – 13. – P. 841 – 850.
4. Wei H., Wang Y. c\ast -Supplemented subgroups and p-nilpotency of finite groups // Ukr. Math. J. – 2007. – 59, № 8. –
P. 1011 – 1019.
Received 27.09.13
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
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| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T02:15:01Z |
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| spelling | umjimathkievua-article-19082019-12-05T09:31:14Z A note on $SΦ$-supplemented subgroups про $SΦ$-доповненi пiдгрупи Li, C. Yi, X. Zhang, Xirong Лі, К. Ій, Х. Чжан, Хіронг We give new and brief proofs of the results obtained by X. Li and T. Zhao in [\mathrm{S}\Phi -supplemented subgroups of finite groups // Ukr. Math. J. – 2012. – 64, № 1. – P. 102–109]. Наведено новi та короткi доведення результатiв, що були отриманi Х. Лi та Т. Жао в [\mathrm{S}\Phi -supplemented subgroups of finite groups // Ukr. Math. J. – 2012. – 64, № 1. – P. 102 – 109]. Institute of Mathematics, NAS of Ukraine 2016-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1908 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 8 (2016); 1139-1141 Український математичний журнал; Том 68 № 8 (2016); 1139-1141 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1908/890 Copyright (c) 2016 Li C.; Yi X.; Zhang Xirong |
| spellingShingle | Li, C. Yi, X. Zhang, Xirong Лі, К. Ій, Х. Чжан, Хіронг A note on $SΦ$-supplemented subgroups |
| title | A note on $SΦ$-supplemented subgroups |
| title_alt | про $SΦ$-доповненi пiдгрупи |
| title_full | A note on $SΦ$-supplemented subgroups |
| title_fullStr | A note on $SΦ$-supplemented subgroups |
| title_full_unstemmed | A note on $SΦ$-supplemented subgroups |
| title_short | A note on $SΦ$-supplemented subgroups |
| title_sort | note on $sφ$-supplemented subgroups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1908 |
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