Prove S4 is not isomorphic to D12SolutionD12 has elements of

Prove S4 is not isomorphic to D12.

Solution

D12 has elements of order 12 (namely r₃₀),

But the maximunm order of an element in S4 is 4.

Since the order of an element is preserved under isomorphism, there can\'t be any isomorphism between these two groups even though both groups have the same number of elements, 4! = 24 = 12 * 2.