Let that (G,∗) be a group of order∣G∣=p , where p is prime. Let a∈G and a=e where e is the identity element of (G,∗). According to Lagrange's theorem, the order of cyclic subgroup ⟨a⟩ divides ∣G∣=p. Therefore, ∣⟨a⟩∣∈{1,p}. Taking into account that the identity e is a unique element of order 1, we conclude that ∣⟨a⟩∣>1, and therefore, ∣⟨a⟩∣=p. Since ⟨a⟩⊆G and ∣⟨a⟩∣=p=∣G∣, we conclude that ⟨a⟩=G. Consequently, G is a cyclic group generated by a.
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