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 John Armstrong | Algebra, Group Homomorphisms, Group theory

10 Comments »

[…] Okay, it’s been pointed out to me that what I was thinking of in my update to yesterday’s post was a little more general than group theory. In the case of groups, preserving the composition is […]

Pingback by Group homomorphisms erratum « The Unapologetic Mathematician | February 12, 2007 | Reply
[…] spend a bit more time on: “images” and “kernels”. Let’s consider a homomorphism […]

Pingback by Subgroups « The Unapologetic Mathematician | February 25, 2007 | Reply
Unlee I am missing something, f3 is not an isomorphism.

For it to be an isomorphism, you need to change the codomain from R* to R+*, i.e. nonzero positive real numbers, with multiplication as composition.

Comment by Fabien | February 28, 2007 | Reply
Ouch. I think I broke that when I recently went through to add TeX. Thanks for catching that.

Comment by John Armstrong | February 28, 2007 | Reply
[…] There is a special kind of function between rings, just like we have in groups. Given rings and , a function is called a homomorphism if it preserves all the ring […]

Pingback by Ring homomorphisms « The Unapologetic Mathematician | March 31, 2007 | Reply
[…] a logarithm because it satisfies the “logarithmic property”. Simply put, it’s a homomorphism of groups from the group of positive real numbers under multiplication to the group of all real numbers under […]

Pingback by The Logarithmic Property « The Unapologetic Mathematician | April 15, 2008 | Reply
Update to your update: let f be a homomorphism G->G’, let e be the identity in G and e’ the identity in G’, and let a be in G. Then f(a)=f(ae)=f(a)f(e); premultiplying by the inverse of f(a) in G’ gives e’ = f(e). No need for the extra requirement.

Comment by Tom S | April 15, 2010 | Reply
There are at least two different ways of composing permutations. The differences turn out not to be deep, but it is very helpful to know which method is being used. Could you describe your method of composing permutations? Here is an example from your text: (bc)(ab) = (acb)… how do you arrive at the answer (acb)? Thank you.

Comment by dratman | April 27, 2012 | Reply
I compose them right-to-left, like functions.

Comment by John Armstrong | April 27, 2012 | Reply
[…] https://unapologetic.wordpress.com/2007/02/10/group-homomorphisms/ […]

Pingback by AUTOMORPHISM for 12 year olds | Peter's ruminations | August 25, 2012 | Reply

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