On finite core-\(p^{2}\) \(p\)-groups
For a positive integer $n$, a group $G$ is said to be a core-$n$ group if $|H : H_G| \leq $n for every subgroup $H$ of $G$, where $H_G$ is the core of $H$ in $G$, i.e., the intersection of all $H^g$ ($g\in G$). It is known that if $G$ is a finite core-$p^k$ $p$-group, where $k$ is a positive integer, then the nilpotency class of $G$ is at most $(k + 1)(k^2 + k + 2)+1$. In [J. Algebra 188, No. 2, 701--719 (1997; Zbl 0878.20014)], \textit{G. Cutolo} et al. studied finite core-$p$ $p$-groups. \par In the paper under review, the authors study finite core-$p^2$ $p$-groups. They prove that for such groups, if $p\geq 3$, then $\mho_{2}\leq Z(G)$, $\mho_{1}(G)$ is abelian, and the nilpotent class of $G$ is at most 5, the derived length is at most 3, and the latter two bounds are best possible. (The case $p=2$ seems to be hard and is not considered here.)
scientific article; zbMATH DE number 6865847