For any prime p show that Cp2 and Cp Cp are not isomorphicSo

For any prime p, show that Cp^2 and Cp ×Cp are not isomorphic.

Solution

We can prove this using a counter example

We know that for Cp² to be isomorphic to CpxCp, the order of elements must be same. Let us see a counter example:

Identity element of C₄ is of order 4. However, no element of C₂xC₂ has order 4. Therefore, the two groups cannot be isomorphic.

Therefore, by counter example, Cp^2 and CpxCp are not isomorphic.