We say that a subgroup $H$ is isolated in a group $G$ if for every $x\in G$ we have either $x\in H$ or $\langle x\rangle\cap H=1$. In this short note, we describe the set of isolated subgroups of a finite abelian group. The technique used is based on an interesting connection between isolated subgroups and the function sum of element orders of a finite group.
On a bijection between a finite group to a non-cyclic group with divisibility of element orders
The Number of Finite Groups Whose Element Orders is Given
The solvability of a finite group by the sum of powers of element orders
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