Cyclicity is 2-local for finite groups

Statement
Suppose $$G$$ is a finite group with the property that for any two elements $$x,y \in G$$, the subgroup $$\langle x,y \rangle$$ is a fact about::cyclic group. Then, $$G$$ itself is a cyclic group.

The proof generalizes to $$G$$ a fact about::finitely generated group.

Related facts

 * Abelianness is 2-local
 * Solvability is 2-local for finite groups
 * Nilpotency is 2-local for finite groups

Proof
Given: A finite group $$G$$ with the property that for any $$x,y \in G$$, the subgroup $$\langle x,y \rangle$$ is cyclic.

To prove: $$G$$ is cyclic.

Proof: Let $$A = \{ a_1, a_2, \dots, a_r \}$$ be a generating set of minimum cardinality for $$G$$ (such a generating set exists because $$G$$ is finite).

Now, if $$r \ge 2$$, consider the subgroup $$\langle a_1, a_2 \rangle$$. By hypothesis, there exists $$b$$ such that $$\langle a_1,a_2 \rangle$$ is generated by $$b$$. Then, the set $$B = \{ b, a_3, \dots, a_r \}$$ also generates $$G$$. Thus, there is a set of size $$r - 1$$ that generates $$G$$, contradicting the minimality of $$r$$.

Thus, $$r = 0$$ or $$r = 1$$. In either case, $$G$$ is cyclic.