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.

Analyzing categorical data
asked 2020-12-25
Suppose that |G| = 24 and that G is cyclic. If a8 e and a12 e, show that = G.

Answers (1)

Given that G is a cyclic group of order 24 and a,b∈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(8,12)=24. Therefore, G=⟨ab⟩.

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