Question

Prove that the groups S₈ and Z₂ X A₈ are not isomorphic. Please provide a proof or explanation of this statement without giving the answer directly.

Answer 1

To prove that S₈ and Z₂ X A₈ are not isomorphic, we compare their group structures and properties, such as order of elements and existence of subgroups.

Step-by-step explanation:

In order to prove that the groups S₈ and Z₂ X A₈ are not isomorphic, we need to show that they do not have the same group structure. To do this, we can compare their properties, such as the order of elements and the existence of certain subgroups.

For example, the symmetric group S₈ has 8! = 40,320 elements, while the direct product Z₂ X A₈ has (2 x 8!) = 80,640 elements. Since the groups have different orders, they cannot be isomorphic.

Additionally, the cyclic subgroup of order 8 in S₈ is isomorphic to Z₈, while there is no cyclic subgroup of order 8 in Z₂ X A₈. Therefore, the two groups do not have the same subgroup structure and are not isomorphic.