External Direct Product of Groups Example: Is ℤ2 ⊕ ℤ2 a Cyclic Group?
The external direct product ℤ2 ⊕ ℤ2 (also written as a Cartesian product ℤ2 x ℤ2) is a non-cyclic group of order 4. We list the elements and make the Cayley table to see that all the non-identity elements have order 2, so none of them is a generator. Therefore, ℤ2 ⊕ ℤ2 is not cyclic. It is in the isomorphism class of the Klein four-group.
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