Isomorphisms for Groups

An isomorphism is a homomorphism that is also a bijection. If there is an isomorphism between two groups G and H, then they are equivalent and we say they are "isomorphic." The groups may look different from each other, but their group properties will be the same.


In mathematics, when you are working with a structure (a group, ring, field, topological space, manifold, etc.) it is key to be able to identify when two structures are identical in their mathematical behavior, even though they may look quite different from one another.


For example, the complex solutions to the equation x^6 - 1=0 form a group under multiplication. That is, if you were to solve this equation and find all 6 roots, you would get 6 complex numbers. These complex numbers form a group.  Also, the group of integers mod 6 form a group under addition.  The group of roots form a group under multiplication, whereas the integers mod 6 form a group under addition.  Believe it or not, these groups are the same.  They are completely identical in their behavior and structure!

We can make precise this idea of “identical structure” by using an isomorphism.  This function clearly identifies which elements in group A correspond to which elements in group B. If this function is a bijection (one-to-one and onto) and preserves the group operations, then you have an isomorphism.

This idea of an isomorphism rears its head all throughout mathematics.  In topology, for example, it’s called a homeomorphism. But in the world of topology, you are comparing two topological spaces instead of two groups.

This video covers:
▪ the distinction between homomorphisms and isomorphisms
▪ several examples of isomorphisms
▪ the memorable derivation of the name “isomorphism”
Isomorphism Example #1
In this example, we show that the real numbers under addition are isomorphic to the group of positive real numbers under multiplication.