Some conditions for cyclic chief factors of finite groups
A subgroup $H$ of a finite group $G$ is called $\scrM$ -supplemented in $G$ if there exists a subgroup $B$ of $G$ such that $G = HB$ and $H_1B$ is a proper subgroup of $G$ for every maximal subgroup $H_1$ of $H$. The main purpose of the paper is to study the influence of $\scrM$ -supplemented subgro...
Saved in:
| Date: | 2016 |
|---|---|
| Main Authors: | , , , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2016
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1957 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507856915136512 |
|---|---|
| author | Gao, B. Miao, L. Tang, J. Гао, Б. Мяо, Л. Тан, Й. |
| author_facet | Gao, B. Miao, L. Tang, J. Гао, Б. Мяо, Л. Тан, Й. |
| author_sort | Gao, B. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:32:42Z |
| description | A subgroup $H$ of a finite group $G$ is called $\scrM$ -supplemented in $G$ if there exists a subgroup $B$ of $G$ such that $G = HB$
and $H_1B$ is a proper subgroup of $G$ for every maximal subgroup $H_1$ of $H$. The main purpose of the paper is to study the
influence of $\scrM$ -supplemented subgroups on the cyclic chief factors of finite groups.
|
| first_indexed | 2026-03-24T02:15:58Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 512.5
J. Tang (Wuxi Inst. Technology, China),
L. Miao (School Math. Sci., Yangzhou Univ., China),
B. Gao (School Math. and Statistics, Yili Normal Univ., China)
SOME CONDITIONS FOR CYCLIC CHIEF FACTORS OF FINITE GROUPS*
ДЕЯКI УМОВИ НА ЦИКЛIЧНI ГОЛОВНI ФАКТОРИ СКIНЧЕННИХ ГРУП
A subgroup H of a finite group G is called \scrM -supplemented in G if there exists a subgroup B of G such that G = HB
and H1B is a proper subgroup of G for every maximal subgroup H1 of H. The main purpose of the paper is to study the
influence of \scrM -supplemented subgroups on the cyclic chief factors of finite groups.
Пiдгрупа H скiнченної групи G називається \scrM -доповненою в G, якщо iснує пiдгрупа B групи G така, що
G = HB, а H1B є власною пiдгрупою G для кожної максимальної пiдгрупи H1 в H. Основною метою статтi є
вивчення впливу \scrM -доповнених пiдгруп на циклiчнi головнi фактори скiнченних груп.
1. Introduction. All groups in this paper are finite. Most of the notation is standard and can be
found in [2, 7, 8]. In what follows, \scrU denotes the formation of all supersoluble groups and \scrN denotes
the formation of all nilpotent groups. The symbol \scrA (p - 1) [12] stands for the formation of all
Abelian groups of exponent dividing p - 1 where p is a prime. F \ast (E) stands for the generalized
Fitting subgroup of E, which coincides with the product of all normal quasinilpotent subgroups of E
[8] (Chapter X). Following Doerk and Hawkes [2], we use [A]B to denote the semidirect product of
the groups A and B, where B is an operator group of A. Z\scrU (G) is the product of all such normal
subgroups H of G whose G-chief factors are cyclic [2].
Let \scrF be a class of groups. If 1 \in \scrF , then we write G\scrF to denote the intersection of all normal
subgroups N of a group G with G/N \in \scrF . The class \scrF is said to be a formation if either \scrF = \varnothing
or 1 \in \scrF and every homomorphic image of G/G\scrF belongs to \scrF for any group G. The formation \scrF
is said to be solubly saturated if G \in \scrF whenever G/\Phi (N) \in \scrF for some soluble normal subgroup
N of a group G.
In this paper, as a continuation of the Theorem of [10], we mainly prove the following theorem.
Theorem 1.1. Let X = E or X = F \ast (E) be two normal subgroups of a group G. Suppose
that every noncyclic Sylow subgroup P of X has a subgroup D such that 1 < | D| < | P | and every
subgroup H of P with order | D| is \scrM -supplemented in G, then each chief factor of G below E is
cyclic.
* This research is supported by NSFC (G.11271016 and G.11501235), the Scientific Research Foundation of Wuxi
Institute of Technology (No. 4016011931), Qing Lan project of Jiangsu Province and High-level personnel of support
program of Yangzhou University.
c\bigcirc J. TANG, L. MIAO, B. GAO, 2016
1718 ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
SOME CONDITIONS FOR CYCLIC CHIEF FACTORS OF FINITE GROUPS 1719
Definition 1.1. A subgroup H is called \scrM -supplemented in a group G, if there exists a sub-
group B of G such that G = HB and H1B is a proper subgroup of G for every maximal
subgroup H1 of H.
Recall that a subgroup H is called weakly S -permutable in a group G [10], if there exists a
subnormal subgroup K of G such that G = HK and H \cap K \leq HsG. In fact, the following example
indicates that the \scrM -supplementation of subgroups cannot be deduced from Skiba’s result.
Example 1.1. Let G = S4 be the symmetric group of degree 4 and H = \langle (1234)\rangle be a cyclic
subgroup of order 4. Then G = HA4 where A4 is the alternating group of degree 4. Obviously, H
is \scrM -supplemented in G because the group H has an unique maximal subgroup. On the other hand,
we have HsG = 1. Otherwise, if H is S -permutable in G, then H is normal in G, a contradiction.
If HsG = \langle (13)(24)\rangle is S -permutable in G, then \langle (13)(24)\rangle is normal in G, also is a contradiction.
Therefore H is not weakly S -permutable in G.
2. Proof of Theorem 1.1. In order to prove Theorem 1.1, we first list here some lemmas.
Lemma 2.1 ([9], Lemmas 2.1 and 2.2). Let G be a group. Then the following hold:
(1) If H \leq M \leq G and H is \scrM -supplemented in G, then H is also \scrM -supplemented in M.
(2) Let N \trianglelefteq G and N \leq H \leq G. If H is \scrM -supplemented in G, then H/N is \scrM -sup-
plemented in G/N.
(3) Let K be a normal \pi \prime -subgroup and H be a \pi -subgroup of G for a set \pi of primes. Then
H is \scrM -supplemented in G if and only if HK/K is \scrM -supplemented in G/K.
(4) If P is a p-subgroup of G where p \in \pi (G) and P is \scrM -supplemented in G, then there
exists a subgroup B of G such that P \cap B = P1 \cap B = \Phi (P ) \cap B and | G : P1B| = p for every
maximal subgroup P1 of P.
Lemma 2.2 ([3], Theorem 1.8.17). Let N be a nontrivial soluble normal subgroup of a group G.
If N \cap \Phi (G) = 1, then the Fitting subgroup F (N) of N is the direct product of minimal normal
subgroups of G which are contained in N.
Lemma 2.3 ([10], Lemma 1). Given a normal p-subgroup E of a group G, if E \leq Z\scrU (G),
then (G/CG(E))\scrA (p - 1) \leq Op(G/CG(E)).
Lemma 2.4 ([10], Lemma 2). Given a normal subgroup E of a group G, if every G-chief factor
of F \ast (E) is cyclic, then so is every G-chief factor of E.
Lemma 2.5 ([1], Lemma 3.5). Let P be a normal p-subgroup of a group G. If each subgroup
of P of order p is complemented in G, then P \leq Z\scrU (G).
Proof of Theorem 1.1. Suppose that this theorem is false and consider a counterexample (G,E)
for which | G| | E| is minimal. Take a Sylow p-subgroup P of E, where p is the smallest prime
divisor of the order of E, and put C = CG(P ). If X = E, then we have following claims.
(1) E is supersoluble and E \not = G.
Corollary 3.3 of [9] shows that E is supersoluble and hence E \not = G by the choice of G.
(2) If T is a Hall subgroup of E, then the hypotheses of Theorem 1.1 hold for (T, T ). Further-
more, if T is normal in G, then the hypotheses of the theorem hold for (G,T ) and (G/T,E/T ).
The claim follows directly from Lemma 2.1.
(3) If T is a nontrivial normal Hall subgroup of E, then T = E.
Suppose that 1 \not = T \not = E. Since T is a characteristic subgroup of E, it follows that T is
normal in G, and by (2) the hypotheses of Theorem 1.1 hold for (G/T,E/T ) and (G,T ). Then
E/T \leq Z\scrU (G/T ) and T \leq Z\scrU (G) by the choice of (G,E). So, E \leq Z\scrU (G). This contradiction
shows that T = E.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
1720 J. TANG, L. MIAO, B. GAO
(4) E = P.
Suppose that E \not = P. By (1), there exists a normal Hall p\prime -subgroup V of E and 1 \not = V \not = E,
which contradicts (3). Consequently, E = P and P is noncyclic.
(5) | D| > p.
Suppose that | D| = p. Then every minimal subgroup of P is \scrM -supplemented in G. Indeed,
every minimal subgroup of P is complemented in G and so by Lemma 2.5, E \leq Z\scrU (G), a
contradiction.
(6) | N | \leq | D| for any minimal normal subgroup N of G contained in E.
Assume that | D| < | N | . Suppose H < N and H is a subgroup of N with order | D| . By
hypotheses, there exists a subgroup B of G such that G = HB and H1B < G for every maximal
subgroup H1 of H. Clearly, G = HB = NB and N \cap B \trianglelefteq G. If N \cap B = N, then G = B, a
contradiction. If N \cap B = 1, then H = N, also is a contradiction.
(7) If N is a minimal normal subgroup of G contained in E, then the hypotheses are still true
for (G/N,E/N).
If | D| = | N | , then N is \scrM -supplemented in G. There exists a subgroup B of G such
that G = NB and TB < G for every maximal subgroup T of N. Clearly, N \nleq TB and | G :
TB| = p. Hence | N | = p, contrary to (5). So we may assume that | N | < | D| , then every subgroup
H/N of P/N with order | D| /| N | is \scrM -supplemented in G/N by Lemma 2.1(2). It follows that
the hypotheses are still true for (G/N,E/N).
(8) P \cap \Phi (G) \not = 1.
If P \cap \Phi (G) = 1, then by Lemma 2.2, P = R1 \times . . . \times Rt with minimal normal subgroups
R1, ..., Rt of G contained in P. Let L be any minimal normal subgroup of G contained in P. We
get | D| \geq | L| by (6). Now we suppose that L \leq H \leq P with | H| = | D| . By hypotheses, there
exists B \leq G such that G = HB and HiB < G for every maximal subgroup Hi of H. Since | G :
HiB| = p by Lemma 2.1(4) and P \cap \Phi (G) = 1, there exists a maximal subgroup Hi of H with
L \nleq Hi and hence H = LHi as well as G = HB = LHiB and L \cap HiB \trianglelefteq G. As L is minimal
normal in G, we get L \nleq HiB and thus | L| = | G : HiB| = p, otherwise, if L \leq HiB, then
HiB = LHiB = HB = G, a contradiction. By hypotheses and (7), (G/L,E/L) satisfies the
condition of Theorem 1.1. The minimal choice of G implies that E/L \leq Z\scrU (G/L) and hence
E \leq Z\scrU (G), a contradiction.
(9) \Phi (P ) \not = 1.
By (8), P \cap \Phi (G) \not = 1. Then there exists a minimal normal subgroup L of G contained in
P \cap \Phi (G) and L is an elementary Abelian p-group.
If | D| = | L| , then we may choose a subgroup H \leq L. By hypotheses, H is \scrM -supplemented
in G, i.e., there exists a subgroup B of G such that G = HB and TB < G for every maximal
subgroup T of H. Since L \leq \Phi (G), we get G = HB = LB = B, a contradiction.
So we have | D| > | L| and fix H \leq P with L < H where | H| = | D| . By hypotheses, H is \scrM -
supplemented in G, i.e., there exists a subgroup B of G such that G = HB and TB < G for every
maximal subgroup T of H. By Lemma 2.1(4), | G : TB| = p and H \cap B = T \cap B \leq \Phi (H) \leq \Phi (P ).
Since L is a minimal normal subgroup of G and TB is a maximal subgroup of G for every maximal
subgroup T of H, we have G = LTB or L \leq TB. If G = LTB for some maximal subgroup T
of H, we obtain G = TB since L is contained in P \cap \Phi (G), a contradiction. Therefore L \leq TB
for every maximal subgroup T of H. Moreover, if L \nleq Ti for some maximal subgroup Ti of H,
then H = LTi and hence TiB = LTiB = HB = G, a contradiction. Therefore we have L \leq T for
every maximal subgroup T of H and hence L \leq \Phi (H) \leq \Phi (P ), that is, \Phi (P ) \not = 1.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
SOME CONDITIONS FOR CYCLIC CHIEF FACTORS OF FINITE GROUPS 1721
(10) CG(P/\Phi (P ))/C is a p-group.
Firstly, we obtain \Phi (P ) \not = 1 by (9). And then we suppose that this claim is false. Pick a p\prime -element
aC of CG(P/\Phi (P ))/C, where a \in CG(P/\Phi (P ))\setminus C. Put G0 = [P ](G/C). Then aC is a nontrivial
p\prime -element of G/C, that is, aC is a p\prime -automorphism of the p-group P and aC \in CG0(P/\Phi (P )),
which contradicts Theorem 1.4 of [6] (Chapter 5). Hence, CG(P/\Phi (P ))/C is a p-group.
(11) P/\Phi (P ) \nleqslant Z\scrU (G/\Phi (P )).
Suppose that P/\Phi (P ) \leq Z\scrU (G/\Phi (P )). Then (G/CG(P/\Phi (P )))\scrA (p - 1) is a p-group by Lemma
2.3. Since
(G/CG(P/\Phi (P )))\scrA (p - 1) \sim = (G/C/CG(P/\Phi (P ))/C)\scrA (p - 1) =
= (G/C)\scrA (p - 1)CG(P/\Phi (P ))/C/CG(P/\Phi (P ))/C,
we get that (G/C)\scrA (p - 1) is a p-group by (10). Take an arbitrary chief factor H/K of G below
\Phi (P ) and C = CG(P ) \leq CG(H/K). Then
(G/CG(H/K))\scrA (p - 1) \sim = (G/C/CG(H/K)/C)\scrA (p - 1) =
= (G/C)\scrA (p - 1)CG(H/K)/C/CG(H/K)/C
and hence (G/CG(H/K))\scrA (p - 1) is a p-group since (G/C)\scrA (p - 1) is a p-group. On the other hand, we
have Op(G/CG(H/K)) = 1 by Lemma 3.9 of [3] (Chapter 1) and then (G/CG(H/K))\scrA (p - 1) = 1.
So G/CG(H/K) \in \scrA (p - 1) and hence | H/K| = p by Lemma 4.1 of [11] (Chapter 1). Therefore
P \leq Z\scrU (G). This contradiction completes the proof of (11).
The final contradiction. It follows form (7), (8), (9) that P/\Phi (P ) \leq Z\scrU (G/\Phi (P )), which
contradicts (11).
If X = F \ast (E), then F \ast (E) \leq Z\scrU (G) by the proof in the case X = E, which by Lemma 2.4
implies that E \leq Z\scrU (G).
Theorem 1.1 is proved.
Note that if \scrF is a solubly saturated formation and G/E \in \scrF , where every chief factor of G
below E is cyclic, then G \in \scrF (Lemma 3.3 in [4]). Therefore from Theorem 1.1 we get the following
corollary.
Corollary 2.1. Let \scrF be a solubly saturated formation containing all supersoluble groups and
X \leq E normal subgroups of a group G such that G/E \in \scrF . Suppose that every noncyclic Sylow
subgroup P of X has a subgroup D such that 1 < | D| < | P | and every subgroup H of P with
order | H| = | D| is \scrM -supplemented in G. If either X = E or X = F \ast (E), then G \in \scrF .
In detail, if \scrF is a saturated formation containing \scrN , then both \scrF \ast and \scrF p
\ast are solubly saturated
formations, where \scrF \ast and \scrF p
\ast denote the class of all quasi-\scrF -groups and the class of all p-quasi-
\scrF -groups, respectively (Theorem A in [5]). Hence we get the following corollary.
Corollary 2.2. Let E be a normal subgroup of a group G such that G/E is p-quasisupersoluble.
Suppose that every noncyclic Sylow subgroup P of X has a subgroup D such that 1 < | D| < | P |
and every subgroup H of P with order | H| = | D| is \scrM -supplemented in G, where X = E or
X = F \ast (E). Then G is p-quasisupersoluble.
References
1. Asaad M. Finite groups with certain subgroups of Sylow subgroups complemented // J. Algebra. – 2010. – 323. –
P. 1958 – 1965.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
1722 J. TANG, L. MIAO, B. GAO
2. Doerk K., Hawkes T. Finite soluble groups. – Berlin; New York: Walter de Gruyter, 1992.
3. Guo W. The theory of classes of groups. – Beijing etc.: Sci. Press-Kluwer Acad. Publ., 2000.
4. Guo W., Skiba A. N. On \scrF \Phi \ast -hypercentral subgroups of finite groups // J. Algebra. – 2012. – 372. – P. 275 – 292.
5. Guo W., Skiba A. N. On some class of finite quasi-\scrF -groups // J. Group Theory. – 2009. – 12. – P. 407 – 417.
6. Gorenstein D. Finite groups. – New York: Chelsea, 1980.
7. Huppert B. Endliche Gruppen I. – Berlin etc.: Springer-Verlag, 1967.
8. Huppert B., Blackburn N. Finite Groups III. – Berlin; New York: Springer-Verlag, 1982.
9. Miao L., Lempken W. On \scrM -supplemented subgroups of finite groups // J. Group Theory. – 2009. – 12, № 2. –
P. 271 – 289.
10. Skiba A. N. Cyclicity conditions for G-chief factors of normal subgroups of a group G // J. Sib. Math. – 2011. – 52,
№ 1. – P. 127 – 130.
11. Shemetkov L. A. Formations of finite groups. – Moscow: Nauka, 1978.
12. Shemetkov L. A., Skiba A. N. Formations of algebraic systems. – Moscow: Nauka, 1989.
Received 17.12.13,
after revision — 01.09.16
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
|
| id | umjimathkievua-article-1957 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:15:58Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d9/4317f684abf5085e6f66028062269bd9.pdf |
| spelling | umjimathkievua-article-19572019-12-05T09:32:42Z Some conditions for cyclic chief factors of finite groups Деякi умови на циклiчнi головнi фактори скiнченних груп Gao, B. Miao, L. Tang, J. Гао, Б. Мяо, Л. Тан, Й. A subgroup $H$ of a finite group $G$ is called $\scrM$ -supplemented in $G$ if there exists a subgroup $B$ of $G$ such that $G = HB$ and $H_1B$ is a proper subgroup of $G$ for every maximal subgroup $H_1$ of $H$. The main purpose of the paper is to study the influence of $\scrM$ -supplemented subgroups on the cyclic chief factors of finite groups. Пiдгрупа $H$ скiнченної групи $G$ називається $\scrM$ -доповненою в $G$, якщо iснує пiдгрупа $B$ групи $G$ така, що $G = HB$, а $H_1B$ є власною пiдгрупою $G$ для кожної максимальної пiдгрупи $H_1$ в $H$. Основною метою статтi є вивчення впливу $\scrM$ -доповнених пiдгруп на циклiчнi головнi фактори скiнченних груп. Institute of Mathematics, NAS of Ukraine 2016-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1957 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 12 (2016); 1718-1722 Український математичний журнал; Том 68 № 12 (2016); 1718-1722 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1957/939 Copyright (c) 2016 Gao B.; Miao L.; Tang J. |
| spellingShingle | Gao, B. Miao, L. Tang, J. Гао, Б. Мяо, Л. Тан, Й. Some conditions for cyclic chief factors of finite groups |
| title | Some conditions for cyclic chief factors of finite groups |
| title_alt | Деякi умови на циклiчнi головнi фактори скiнченних груп |
| title_full | Some conditions for cyclic chief factors of finite groups |
| title_fullStr | Some conditions for cyclic chief factors of finite groups |
| title_full_unstemmed | Some conditions for cyclic chief factors of finite groups |
| title_short | Some conditions for cyclic chief factors of finite groups |
| title_sort | some conditions for cyclic chief factors of finite groups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1957 |
| work_keys_str_mv | AT gaob someconditionsforcyclicchieffactorsoffinitegroups AT miaol someconditionsforcyclicchieffactorsoffinitegroups AT tangj someconditionsforcyclicchieffactorsoffinitegroups AT gaob someconditionsforcyclicchieffactorsoffinitegroups AT mâol someconditionsforcyclicchieffactorsoffinitegroups AT tanj someconditionsforcyclicchieffactorsoffinitegroups AT gaob deâkiumovinacikličnigolovnifaktoriskinčennihgrup AT miaol deâkiumovinacikličnigolovnifaktoriskinčennihgrup AT tangj deâkiumovinacikličnigolovnifaktoriskinčennihgrup AT gaob deâkiumovinacikličnigolovnifaktoriskinčennihgrup AT mâol deâkiumovinacikličnigolovnifaktoriskinčennihgrup AT tanj deâkiumovinacikličnigolovnifaktoriskinčennihgrup |