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On p-soluble groups with a generalized p-central or powerful sylow p-subgroup | ||
| International Journal of Group Theory | ||
| مقاله 5، دوره 1، شماره 2، شهریور 2012، صفحه 51-57 اصل مقاله (413.52 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2012.761 | ||
| نویسنده | ||
| Evgeny Khukhro* | ||
| Sobolev Institute of Mathematics, Novosibirsk | ||
| چکیده | ||
| Let $G$ be a finite $p$-soluble group, and $P$ a Sylow $p$-subgroup of $G$. It is proved that if all elements of $P$ of order $p$ (or of order ${}\leq 4$ for $p=2$) are contained in the $k$-th term of the upper central series of $P$, then the $p$-length of $G$ is at most $2m+1$, where $m$ is the greatest integer such that $p^m-p^{m-1}\leq k$, and the exponent of the image of $P$ in $G/O_{p',p}(G)$ is at most $p^m$. It is also proved that if $P$ is a powerful $p$-group, then the $p$-length of $G$ is equal to 1. | ||
| کلیدواژهها | ||
| $p$-central $p$-group of height $k$؛ powerful $p$-group؛ $p$-soluble؛ $p$4-length | ||
| مراجع | ||
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E. G. Bryukhanova (1979) The 2-length and 2-period of a finite solvable group Algebra Logika 18, 9-31
D. Gorenstein (1980) Finite groups Chelsea, New York
F. Gross (1965) The 2-length of a finite solvable group Pacif. J. Math. 15, 1221-1237
J. Gonz\'alez-S\'anchez and T. S. Weigel (2011) Finite $p$-central groups of height $k$ Isr. J. Math. 181, 125-143
P. Hall and G. Higman (1956) The $p$-length of a $p$-soluble group and reduction
theorems for Burnside's problem Proc. London Math. Soc. (3) 6, 1-42
B. Huppert (1967) Endliche Gruppen. I,
Springer, Berlin
A. Lubotzky and A. Mann (1987) Powerful $p$-groups. I: Finite groups J. Algebra 105, 484-505
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