# Suppose that |G| = 24 and that G is cyclic. If a8 e and a12 e, show that = G.

Suppose that $|G|=24$ and that G is cyclic. If a8 e and a12 e, show that = G.

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delilnaT

Given that G is a cyclic group of order 24 and a,$b\in G$ has order 8 and 12, respectively.

Now to show $G=⟨ab⟩$.

Since, G is cyclic, so it is abelian, So aa commutes with b. Therefore, order of $ab=lcm$(order of a, order of b)= $lcm\left(8,12\right)=24$.

Therefore, $G=⟨ab⟩$.