Isomorphic groups have the same structure - they describe the same type of symmetry. You can think of two isomorphic groups as two different realizations of the same abstract group. For example, the two C2 groups of {0 degrees, 180 degrees}, composition of consecutive rotation, and {identity transformation, mirror reflection}, composition of consecutive reflection, are really the same group. You can describe an isosceles triangle by either its mirror symmetry or rotational symmetry (about the axis in the plane), but you will still be describing the same isosceles triangle.
One way to distinguish isomorphic groups is that they have the same group table. If you can rearrange the rows and columns or re-denote elements so that the group tables are identical, then the two groups are isomorphic.
Page author: Sasie Sealy