Final answer:
S4 and D12 cannot be isomorphic because they have elements of different maximum orders; S4 has elements of order up to 4 while D12 has elements of order 6.
Step-by-step explanation:
To prove that the symmetric group on 4 elements, denoted as S4, is not isomorphic to the dihedral group of order 12, denoted as D12, we can examine properties that are preserved under isomorphisms and show a discrepancy between the two groups.
S4 consists of all the permutations of four elements and has a total of 4! = 24 elements. D12, on the other hand, represents the symmetries of a regular hexagon and has exactly 12 elements, which include 6 rotations and 6 reflections.
One property to look at is the number of elements of a particular order. In S4, the maximum order of an element (which is the order of a 4-cycle) is 4. However, in D12, the maximum order of an element (which is the order of a 180-degree rotation) is 6. Since isomorphisms preserve the order of elements, the two groups cannot be isomorphic due to this difference in the order of elements.
Another property is the center of the group, which consists of elements that commute with every other element in the group. The center of S4 is trivial; it only contains the identity element. D12 also has a trivial center. While this particular property does not distinguish the two groups, it's an example of how one would compare group properties in the context of isomorphism.
Thus, with the evident difference in the orders of elements, specifically that no element in S4 has an order of 6 while an element in D12 does, we can conclude that S4 is not isomorphic to D12.