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G. Cutolo, E.I. Khukhro, J.C. Lennox, J. Wiegold, S. Rinauro and H. Smith
Finite core-p p-groups
J. Algebra, 188, (1997), pp. 701–719.
doi: 10.1006/jabr.1996.6811
MathSciNet Zentralblatt Comments Abstract Full Text
Abstract
For $n$ a positive integer, a group $G$ is called core-$n$ if $H/H_G$ has order at most $n$ for every subgroup $H$ of $G$ (where $H_G$ is the normal core of $H$, the largest normal subgroup of $G$ contained in $H$). It is proved that a finite core-$p$ $p$-group $G$ has a normal abelian subgroup whose index in $G$ is at most $p^2$ if $p\ne 2$, which is the best possible bound, and at most $2^6$ if $p = 2$.
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