Yes, two groups can have the same number of elements but still not be isomorphic. This is because isomorphism not only considers the number of elements, but also the structure and operations of the groups.
An embedding is a function that preserves the structure and operation of a group, whereas an isomorphism is a bijective embedding that also preserves the structure and operation of the group.
Yes, a group can be embedded into itself through the identity function, where each element is mapped to itself. This is known as the trivial embedding.
To show that there is no embedding between two groups, we can look for structural differences between the two groups. If there are elements or operations in one group that do not exist in the other, then an embedding cannot exist.
Yes, there are groups that cannot be embedded into each other. For example, the cyclic group of order 4 cannot be embedded into the symmetric group of order 3, as the symmetric group does not have an element of order 4.