7. We say a finite group G is Lagrangian if it has a subgroup of order d for any d | |G|. For example: 1 • We proved that if G is a finite cyclic group, then G has exactly one subgroup of order d for each d| |G|. Hence every finite cyclic group is Lagrangian. Every finite p-group is Lagrangian by the first Sylow theorem. A4 is not Lagrangian since it has no subgroup of order 6. Prove the following statements. (a) If G₁ and G₂ are Lagrangian, then so is G₁ × G₂. (b) Every finite abelian group is Lagrangian. ●

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