Explain Is D12 isomorphic to S4 Is D4 times D4 isomorphic to

Explain.

Is D_12 isomorphic to S_4? Is D_4 times D_4 isomorphic to C_4 times C_4 times C_4? Explain.

Solution

D12 has an element of order 12. But the maximum order of an element in S4 is 4.

Since the order of an element is preserved under isomorphism, There cannot be any isomorphism between these two groups even though both groups have the same number of elements 4!=24=12*2

Thus D12 is not isomorphic to S4