| Title:
|
On $S$-quasinormal and $c$-normal subgroups of a finite group (English) |
| Author:
|
Li, Shirong |
| Author:
|
Li, Yangming |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
58 |
| Issue:
|
4 |
| Year:
|
2008 |
| Pages:
|
1083-1095 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\cal F$ be a saturated formation containing the class of supersolvable groups and let $G$ be a finite group. The following theorems are presented: (1) $G\in \cal F$ if and only if there is a normal subgroup $H$ such that $G/H\in \cal F$ and every maximal subgroup of all Sylow subgroups of $H$ is either $c$-normal or $S$-quasinormally embedded in $G$. (2) $G\in \cal F$ if and only if there is a normal subgroup $H$ such that $G/H\in \cal F$ and every maximal subgroup of all Sylow subgroups of $F^*(H)$, the generalized Fitting subgroup of $H$, is either $c$-normal or $S$-quasinormally embedded in $G$. (3) $G\in \cal F$ if and only if there is a normal subgroup $H$ such that $G/H\in \cal F$ and every cyclic subgroup of $F^*(H)$ of prime order or order 4 is either $c$-normal or $S$-quasinormally embedded in $G$. (English) |
| Keyword:
|
$S$-quasinormally embedded subgroup |
| Keyword:
|
$c$-normal subgroup |
| Keyword:
|
$p$-nilpotent group |
| Keyword:
|
the generalized Fitting subgroup |
| Keyword:
|
saturated formation |
| MSC:
|
20D10 |
| MSC:
|
20D20 |
| MSC:
|
20D40 |
| MSC:
|
20E28 |
| idZBL:
|
Zbl 1166.20013 |
| idMR:
|
MR2471167 |
| . |
| Date available:
|
2010-07-21T08:10:58Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140441 |
| . |
| Reference:
|
[1] Asaad, M., Heliel, A. A.: On $S$-quasinormal embedded subgroups of finite groups.J. Pure App. Algebra 165 (2001), 129-135. MR 1865961, 10.1016/S0022-4049(00)00183-3 |
| Reference:
|
[2] Ballester-Bolinches, A., Pedraza-Aguilera, M. C.: Sufficient conditions for supersolvability of finite groups.J. Pure App. Algebra 127 (1998), 113-118. MR 1620696, 10.1016/S0022-4049(96)00172-7 |
| Reference:
|
[3] Deskins, W. E.: On quasinormal subgroups of finite groups.Math. Z. 82 (1963), 125-132. Zbl 0114.02004, MR 0153738, 10.1007/BF01111801 |
| Reference:
|
[4] Kegel, O. H.: Sylow Gruppen und subnormalteiler endlicher Gruppen.Math. Z. 78 (1962), 205-221. Zbl 0102.26802, MR 0147527, 10.1007/BF01195169 |
| Reference:
|
[5] Guo, X. Y., Shum, K. P.: On c-normal maximal and minimal subgroups of Sylow $p$-subgroups of finite groups.Arch. Math. 80 (2003), 561-569. Zbl 1050.20010, MR 1997521, 10.1007/s00013-003-0810-4 |
| Reference:
|
[6] Huppert, B.: Endliche Gruppen I.Springer-Verlag, Berlin-Heidelberg-New York (1967). Zbl 0217.07201, MR 0224703 |
| Reference:
|
[7] Huppert, B., Blackburn, N.: Finite Groups III.Springer-Verlag, Berlin, New York (1982). Zbl 0514.20002, MR 0662826 |
| Reference:
|
[8] Li, D., Guo, X.: The influence of c-normality of subgroups on structure of finite groups.Comm. Algebra 26 (1998), 1913-1922. MR 1621704, 10.1080/00927879808826248 |
| Reference:
|
[9] Li, D., Guo, X.: The influence of c-normality of subgroups on structure of finite groups II.J. Pure App. Algebra 150 (2000), 53-60. MR 1762920, 10.1016/S0022-4049(99)00042-0 |
| Reference:
|
[10] Li, Shirong, He, Xuanli: On normally embedded subgroups of prime power order in finite groups.Comm. Algebra 36 (2008), 2333-2340. Zbl 1146.20015, MR 2418390, 10.1080/00927870701509370 |
| Reference:
|
[11] Li, Yangming, Wang, Yanming: On $\pi$-quasinormally embedded subgroups of finite group.J. Algebra 281 (2004), 109-123. Zbl 1079.20026, MR 2091963, 10.1016/j.jalgebra.2004.06.026 |
| Reference:
|
[12] Schmid, P.: Subgroups permutable with all Sylow subgroups.J. Algebra 207 (1998), 285-293. Zbl 0910.20015, MR 1643106, 10.1006/jabr.1998.7429 |
| Reference:
|
[13] Srinivasan, S.: Two sufficient conditions for supersolvability of finite groups.Israel J. Math. 35 (1980), 210-214. Zbl 0437.20012, MR 0576471, 10.1007/BF02761191 |
| Reference:
|
[14] Tate, J.: Nilpotent quotient groups.Topology 3 (1964), 109-111. Zbl 0125.01503, MR 0160822, 10.1016/0040-9383(64)90008-4 |
| Reference:
|
[15] Wang, Yanming: c-normality of groups and its properties.J. Algebra 180 (1996), 954-965. Zbl 0847.20010, MR 1379219, 10.1006/jabr.1996.0103 |
| Reference:
|
[16] Wei, H.: On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups.Comm. Algebra 29 (2001), 2193-2200. Zbl 0990.20012, MR 1837971, 10.1081/AGB-100002178 |
| Reference:
|
[17] Wei, H., Wang, Y., Li, Y.: On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups II.Comm. Algebra 31 (2003), 4807-4816. Zbl 1050.20011, MR 1998029, 10.1081/AGB-120023133 |
| . |
