= Solution
A <group> $G$ is a <cyclic group> if some element $g\in G$ generates it:
$$
G=\langle g\rangle=\{g^n:n\in\mathbb Z\}.
$$
For a finite <cyclic group>, the <order of a group element> of a generator equals the number of elements of the <group>.
An example of a finite <abelian group> that is not a <cyclic group> is the <Klein four-group>, realized as
$$
G=(\mathbb Z/2\mathbb Z)^2
$$
with componentwise addition modulo two. Addition commutes, so this is an <abelian group> of four elements. Each of its three nonzero elements satisfies $g+g=0$, so the <cyclic subgroup> it generates has just two elements; the identity generates only itself. \b[No element generates all four elements.]
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