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

Document Type : Research Paper

Author

Sobolev Institute of Mathematics, Novosibirsk

Abstract

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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#### References

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