The Unapologetic Mathematician

Mathematics for the interested outsider

Group homomorphisms

At last we come to the notion of a homomorphism of groups. These are really, in my view, the most important parts of the theory. They show up everywhere, and the structure of group theory is intimately bound up with the way homomorphisms work.

So what is a homomorphism? It’s a function from the set of members of one group to the set of members of another that “preserves the composition”. That is, a homomorphism f:G\rightarrow H takes an element g of G and gives back an element f(g) of H. It has the further property that f(g_1g_2)=f(g_1)f(g_2). The product of g_1 and g_2 uses the composition from G, while the product of f(g_1) and f(g_2) uses the composition of H.

Let’s consider an example very explicitly: a homomorphism f1:S_3\rightarrow{\mathbb Z}_2. Remember that S_3 is the group of rearrangements of 3 objects (I’ll use a, b, and c), while {\mathbb Z}_2 is the group of “addition modulo 2”.

S_3 {\mathbb Z}_2
e 0
({\rm b}\,{\rm c}) 1
({\rm a}\,{\rm b}) 1
({\rm a}\,{\rm b}\,{\rm c}) 0
({\rm a}\,{\rm c}) 1
({\rm a}\,{\rm c}\,{\rm b}) 0

If we consider the permutations ({\rm b}\,{\rm c}) and ({\rm a}\,{\rm b}) in S_3, each one is sent to 1 in the group {\mathbb Z}_2, and 1+1 = 0 there. On the other hand, ({\rm b}\,{\rm c})({\rm a},{\rm b})=({\rm a}\,{\rm c}\,{\rm b}), which is sent to 0. The composition of the images is the image of the composition. We can pick any two permutations on the right and see the same thing.

Another example: f_2:{\mathbb Z}\rightarrow{\mathbb Z} with f_2(n)=3n. The homomorphism property says that f_2(m+n)=f_2(m)+f_2(n), and indeed we see that 3(m+n)=3m+3n.

Another: f_3:{\mathbb R}^+\rightarrow{\mathbb R}_+^*. By {\mathbb R}^+. I mean the real numbers with addition as composition, and by {\mathbb R}_+^*. I mean the positive.nonzero real numbers with multiplication. I define f_3(x)=2^x. The laws of exponents tell us that 2^{x+y}=2^x2^y.

As we continue we will see many more examples of homomorphisms. For now, there are a few definitions we will find useful later. Recall from the discussion about functions that a surjection is a function between functions that hits every point in its codomain at least once. A group homomorphism that is also a surjection we call an “epimorphism”. Similarly, an injection is a function that hits every point in its codomain at most once. A group homomorphism that is also an injection we call a “monomorphism”. A homomorphism that is both — the function is a bijection — we call an “isomorphism”. In the above examples, f_1 is an epimorphism, f_2 is a monomorphism, and f_3 is an isomorphism.

If a homomorphism’s domain and codomain group are the same, as in f_2 above, we call it an “endomorphism” on the group. If it’s also an isomorphism we call it an “automorphism”. The homomorphism f_2 is not an automorphism, since it doesn’t hit any point that’s not a multiple of 3.

And finally, a few things to think about.

  • Can you construct a homomorphism from S_n to {\mathbb Z}_2 similar to f_1 above, but for other values of n?
  • What homomorphisms can you construct from {\mathbb Z} to S_3? to S_4? to an arbitrary group G?
  • What homomorphisms can you construct from {\mathbb Z}_3 to S_4?

UPDATE: I just remembered that I left off another technical requirement. A homomorphism has to send the identity of the first group to the identity of the second. It usually doesn’t cause a problem, but I should include it to be thorough. It isn’t hard to verify that all the homomorphisms I mentioned satisfy this property too.

February 10, 2007 Posted by | Algebra, Group Homomorphisms, Group theory | 10 Comments