16.61 (2006)
SolvedA subgroup $H$ of a group $G$ is fully invariant if $\vartheta(H) \leqslant H$ for every endomorphism $\vartheta$ of $G$. Let $G$ be a finite group such that $G$ has a fully invariant subgroup of order $d$ for every $d$ dividing $|G|$. Must $G$ be cyclic?
Progress
No: take $G = C_p \times C_{p^2}$, where $C_p, C_{p^2}$ are cyclic $p$-groups of orders $p, p^2$ (A. Abdollahi, Letter of 4 March 2009).
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