# On intersection of all maximal cyclic subgroups

A maximal cyclic subgroup is a cyclic subgroup that is maximal among all cyclic subgroups. That is there is no other cyclic subgroup that would contain it while there might be other non-cyclic subgroups that are larger.

Why intersection of all maximal cyclic subgroups of a finite group $G$ is contained in the center of $G$ i.e. $Z(G)$?

Let $a \in \cap H$ such that H is maximal cyclic subgroup, and consider $g \in G$ ,$<g>\, \subset H'$ such that $H'$ is maximal cyclic subgroup so $a \in H'$ ,it's means $a,g \in H'$ so $ag=ga$
Let $a$ be an element in the intersection of all maximal cyclic subgroups of $G$. We must show that, for any $g\in G$, $ag=ga$.
This $g$ generates a cyclic subgroup, hence is contained in a maximal cyclic subgroup – as well as $a$. However a cyclic subgroup is commutative, hence $ag=ga$.