Group homomorphisms: definition and examples

 Let (G,*) and (H,@) be groups. A function f:G➔H is a homomorphism if it preserves the operation structure, such that for any two elements g1 and g2 in G, f(g1)@f(g2) = f(g1*g2).

Note that I used different symbols for the group operations of G and H. An additive group can be mapped to a multiplicative group or a group of permutations, and of course, an additive group can be mapped to another additive group.

Corollary #1. Any homomorphism f:G➔H must map the identity element of G to the identity element of H.

 

New definition: The kernel of a homomorphism f:G➔H, denoted as Ker f, is the set of all elements of G that are mapped to the identity element of H. 

Some examples.

f:(R,+)➔(R,+) defined by f(x) = x for all x in R.

 

This kind of mapping can always work from any group to itself, and Ker f = {identity element of G}.

 

f:(R,+)➔(R,+) defined by f(x) = 0 for all x in R.

 

This is another homomorphism that always works from any group G to any group H, and Ker f = G, the entire set.

 

The first two examples are abstract, which is only fair since this is abstract algebra, so let's work with a concrete example you have worked with if you took calculus.

 

Let (P,+) be the group of all polynomials of a single variable under addition. The identity polynomial is f(x) = 0. Let d:(P,+)➔(P,+) be defined as d(f(x)) = f'(x).

 

Differentiation of polynomials is an additive group, because as we learned in calculus f'(x) + g'(x) = (f' + g')(x). Whether you take derivatives first and then add the result, or add polynomials first then take the derivative, the answer will be the same. Ker d = {the set of constant polynomials, f(x) = k, where k is some real number}.

 

One more example, this one from an additive to a multiplicative group.

 

Let exp:(R,+)➔((0,∞),𝗫) be defined as exp(x) = eˣ.

 

Ker exp = {0}, and we know that adding exponents is the same as multiplying their results, exp(x+y) = eˣeʸ.

 

Commentary

 

The options for superscripts when using the Blogger editor are not ideal. All numbers have their superscript symbols, but the letters of Roman alphabet aren't as well represented. You can see in the term eˣeʸ that the x and the y don't line up properly. If I need to write a long passage with exponentiation of variables, I will use Word and import the work as a picture, such as

This is much more aesthetically pleasing.

 

Tomorrow: learning about the most important flavors of homomorphisms, which are monomorphisms, epimorphisms, isomorphisms and automorphisms.