Is there such a group? I just learned that for a non-abelian group $G$, the order of its center $Z$ is at most $1/4$ of the order of $G$, but I can't think of any group for which the equality hold. Could it be that the inequality is strict?
The non-abelian groups of order 8 give you two examples.
The center of the dihedral group of order $8$ is cyclic of order $2$.
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2 years ago