We've looked at groups defined by generators and relations. We've also developed an intuitive notion of what it means for two groups to be the same. This sections will make this concept more precise, placing it in the more general setting of maps between groups.

A *homomorphism* is a map between two groups which respects
the group structure. More formally, let G and H be two group, and f a
map from G to H (for every g∈G, f(g)∈H). Then f is a
homomorphism if for every g_{1},g_{2}∈G,
f(g_{1}g_{2})=f(g_{1})f(g_{2}). For
example, if H<G, then the inclusion map i(h)=h∈G is a
homomorphism. Another example is a homomorphism from Z to Z given by
multiplication by 2, f(n)=2n. This map is a homomorphism since
f(n+m)=2(n+m)=2n+2m=f(n)+f(m).

- Suppose f:G→H is a homomorphism between two groups, with the
identity of G denoted e
_{G}and the identity of H denoted e_{H}. Show that f(e_{G})=e_{H}, that is, identity is sent to identity by any homomorphism. (Hint: use the fact that e=ee and the defining property of homomorphisms.) - Consider the map f:Z
_{9}→Z_{3}given by f(R_{m})=R_{3m}(recall that R_{m}is a counterclockwise rotation by m degrees). Is this a homomorphism? - Find a homomorphism from Z
_{6}to Z_{3}. - Is the map f:Z
_{6}→Z_{5}given by f(R_{m})=R_{0}(the identity) a homomorphism? - Find a homomorphism from F
_{2}to Z×Z. (Hint: why not try sending the generators of one to the generators of the other?)

The last part of the above activity hints at a key fact: a
homomorphism is determined by what elements it sends the generators to.
Suppose the two elements a,b∈G generate G. Then for any
group H and homomorphism f:G→H,
f(aba^{-1}bb)=f(a)f(b)f(a)^{-1}f(b)f(b). Thus we need
only specify f(a) and f(b) to define the homomorphism for every element
in G.

- Prove the equality above by showing that for any homomorphism f and
elements a,b,c, f(abc)=f(a)f(b)f(c) and
f(a
^{-1})=f(a)^{-1}. - Find all homomorphisms from Z to Z and from
F
_{2}to Z_{3}×Z_{3}. (Hint: use generators.)

- Recall that Z
_{3}is generated by R_{120}. Suppose we try to define a homomorphism f:Z_{3}→Z by letting f(R_{120})=1, sending a generator to a generator. Does this extend to a homomorphism? What relation does R_{120}satisfy that 1∈Z does not? - There are many homomorphisms from F
_{2}to Z×Z. Take for instance f(a)=(1,0) and f(b)=(0,2). What are some homomorphisms from Z×Z to F_{2}? (Hint: suppose f is such a homomorphism and f((1,0))=w_{1}, f((0,1))=w_{2}, where w_{1}and w_{2}are words in F_{2}. What relation must w_{1}and w_{2}satisfy?) - If you're itching for a challenge, try to find all the
homomorphisms from Z×Z to F
_{2}. What do they have in common?

Putting the above idea into the language of Cayley graphs, we get that if f:G→H is a homomorphism, and elements a and b generate G, then any loops in the Cayley graph of G with respect to generators a and b must be sent to similarly oriented (possibly trivial) loops in the Cayley graph of H with respect to generators f(a) and f(b). Informally, the Cayley graph of H (with respect to f(a) and f(b)) may have "additional" loops to those in the Cayley graph of G, but it may not have fewer. Here's a picture explanation to make this more clear.

Give a homomorphism f:G→H, the set of all elements h∈H such
that h=f(g) for some g∈G is called the *image* of G and is
denoted f(G). It is the set of elements in H which f maps some element
of G to. If
f(G)=H, we say that f is *surjective* or *onto*.
Similarly, we denote by f^{-1}(h) all the elements in G which f
maps to h. For example, the homomorphism
f:Z_{6}→Z_{3} given by
f(R_{m})=R_{2m} is a surjective homomorphism and
f^{-1}(R_{120})={R_{60},R_{240}}.

- Suppose f:G→H is a homomorphism, e
_{G}and e_{H}the identity elements in G and H respectively. Show that the set f^{-1}(e_{H}) is a subgroup of G. This group is called the*kernel*of f. (Hint: you know that e_{G}∈f^{-1}(e_{H}) from before. Use the definition of a homomorphism and that of a group to check that all the other conditions are satisfied.) - If f
^{-1}(e_{H})={e_{G}}, only the identity of G, how many elements does the set f^{-1}(h) have for any other h∈H? (Hint: suppose v,w∈G and f(v)=f(w)=h. What is f(vw^{-1})? Use the definition of a homomorphism.)

For f:G→H a homomorphism, if f^{-1}(identity) has only
one element—and by problem 2 above we know that this means that f
maps each element of G to a distinct element of H—then we say that
f is *injective* or *one-to-one*. A homomorphism which is
both injective and surjective is called an *isomorphism*, and in
that case G and H are said to be *isomorphic*.

- Find all subgroups of the group D
_{4}. Which of them are isomorphic? Which are normal? (Hint: use the Cayley graph and/or product table to help you.) - Show that for any two groups G and H, the kernel of any homomorphism
f:G→H is a normal subgroup of G. (Hint: Call the kernel K.
Consider the image of g
^{-1}Kg under f, i.e. the set f(g^{-1}Kg) for some g∈G. What does it equal?)

Given a surjective homomorphism f:G→H, let K be it's kernel. Show that the quotient group G/K is isomorphic to H. (Hint: first construct a homomorphism q from G/K to H, and then show that it's surjective and injective. You have only the given homomorphism f to work with, so why not try q(gK)=f(g)? Is this a homomorphism? Is it injective and surjective?)

This theorem is called the first isomorphism theorem, and it will serve as a concluding note in this brief introduction to group theory.