Let ?1 and ?2 be groups. Define the set ?1 ⊕ ?2 ≔ {(?1, ?2) | ?1 ∈ ?1, ?2 ∈ ?2 equipped with operation (?1, ?2) ⋅ (?1, ?2) ≔ (?1?1, ?2?2). Using the definition of a group, show that ?1 ⊕ ?2 is a group with the operation described above. This group is known as the direct sum of ?1 and ?2.

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