On p-soluble groups with a generalized p-central or powerful Sylow p-subgroup

Let $G$ be a finite $p$-soluble group، and $P$ a Sylow $p$-sub-group 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.