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.
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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
Khukhro, E. (2012). On p-soluble groups with a generalized p-central or powerful sylow p-subgroup. International Journal of Group Theory, 1(2), 51-57. doi: 10.22108/ijgt.2012.761
MLA
Khukhro, E. . "On p-soluble groups with a generalized p-central or powerful sylow p-subgroup", International Journal of Group Theory, 1, 2, 2012, 51-57. doi: 10.22108/ijgt.2012.761
HARVARD
Khukhro, E. (2012). 'On p-soluble groups with a generalized p-central or powerful sylow p-subgroup', International Journal of Group Theory, 1(2), pp. 51-57. doi: 10.22108/ijgt.2012.761
CHICAGO
E. Khukhro, "On p-soluble groups with a generalized p-central or powerful sylow p-subgroup," International Journal of Group Theory, 1 2 (2012): 51-57, doi: 10.22108/ijgt.2012.761
VANCOUVER
Khukhro, E. On p-soluble groups with a generalized p-central or powerful sylow p-subgroup. International Journal of Group Theory, 2012; 1(2): 51-57. doi: 10.22108/ijgt.2012.761