Homomorphisms

Recall that an isomorphism f from a group G into a group H is a function f :GH which is injective and which satisfies

for all g, g' G. There are many interesting examples of functions f :GH which satisfy (1) but are not injective. Such functions are fundamental in the theory of groups. Let us make the following definition.

Definition 1: Let G and H be groups. A function f :GH which satisfies (1) for all g, g' G is called a homomorphism of G into H.

Note that the product g · g' is taken with respect to multiplication in G, whereas the product f(g) · f(g') is taken with respect to multiplication in H.

We noted that isomorphic systems are essentially the same, being "abstractly equal". This is certainly not true of homomorphic systems in general. A homomorphic image can be compared to a silhouette of a system projected or mapped onto a screen. Certain features like the profile are preserved, but certain features such as coloration are lost.

Example 1: An isomorphism f :GH is a homomorphism.

Example 2: Let f :GH be defined by

for all g G. Then f is a homomorphism.

Example 3: Let G = D_n, H=[F]. Then every element of D_n is of the form F^aR^b (a = 0,1; b = 0,1,...,n-1). Let us now define f :GH by

To prove that (1) holds, we must show that

However, we showed earlier that

Therefore, the left-hand side of (2) equals F^a+c in both cases, and

which proves (2). Thus, f is a homomorphism.

Example 4: Let f :ZZ be defined by

f(a) = 2a    (a Z).

Since

we see that f is a homomorphism.

Example 5: We may generalize Example 4 as follows: Let G be an abelian group, and let n be an integer. Then let us define f_n :G G by

f_n(a) = aⁿ    (a G).

Since

we see that f_n is a homomorphism.

Proposition 2: let f :GH be a homomorphism.

Proof: (1) Since 1_G· 1_G= 1_G and since f is an isomorphism, we see that

Multiplying both sides by f(1_G)^-1, we get f(1_G) = 1_H.

(2) It suffices to show that

For then by the uniqueness of inverses in H, we have f(a^-1) = f(a)^-1. However,

Therefore, by part (1) and the fact that f is a homomorphism, we get (4).

(3) By Proposition 2 of the section on subgroups, it suffices to show that if a,b K then f(a)· f(b)^-1 f(K).

Suppose that f:GH is a homomorphism. Set

ker(f) = {x G | f(x) = 1_H}.

We will call ker(f) the kernel of f. If F is an isomorphism, then ker(f) = {1_G}. The larger the kernel of f, the farther f is from an isomorphism.

If f:GH is a homomorphism, then ker(f) is a normal subgroup of G.

Proposition 3: If f:GH is a homomorphism, then ker(f) is a normal subgroup of G.

Proof: First let us show that ker(f) is a subgroup of G. Let a,b ker(f). It suffices to show that a · b^-1 ker(f). Now

a,b ker(f) f(a) = f(b) = 1_H

f(a) = f(b^-1) = 1_H (by part (2) of proposition 2 above.)

f(a · b^-1) = f(a) f(b^-1) = 1_H

a · b^-1 ker(f).

Let us now show that ker(f) is normal. Let x ker(f), a G. Then

so that axa^-1 ker(f). Therefore, ker(f) is normal.

If one knows the kernel of a homomorphism, then one has a good deal of information about the homomorphism, as the next two results show.

Let f :GH be a homomorphism, h H. Let g G satisfy f(g) = h. Then f ^-1(h) = g · ker(f).

Proposition 4: Let f :GH be a homomorphism, h H. Let g G satisfy f(g) = h. Then

Proof: If g' g · ker(f), then g' = g · k for some k ker(f). Therefore,

= f(g) [since k ker(f)]

Therefore, g' f ^-1 and g · ker(f) f ^-1(h). In order to prove the reverse inclusion, note that if g' f ^-1(h), then f(g') = h f(g') · f(g)^-1 = h · h^-1 = 1_H. But f(g')·f(g)^-1 = f(g'·g^-1) By Proposition 2 part(2). Therefore,

f(g'·g^-1) = 1_H g'·g^-1 ker(f)

g' g · ker(f)

Thus, f ^-1(h) g · ker(f), and therefore f ^-1(h) = g · ker(f).

Corollary 5: Let f :GH be a homomorphism. Then f is an injection ker(f) = {1_G}.

Proof: clear.

Suppose that ker(f) = {1_G. By Proposition 4, if h f(G), then f ^-1(h) consists of exactly one element. Therefore, f is an injection.

Proposition 3 tells us that with every homomorphism f of G into a group H there is an associated normal subgroup of G, ker(f). Conversely, suppose that we are given a normal subgroup N of G. Then there is an associated homomorphism, which we will now construct. We have already defined the quotient group G/N. Let us associate to N the homomorphism

i_N:GG/N,

Indeed, i_N is a homomorphism since

= i_N(g₁)i_N(g₂)    (g₁,g₂ G)

by the definition of multiplication in G/N

Proposition 6: The homomorphism i_N:GG/N is surjective and ker(i_N) = N.

Proof: It is clear that i_N is surjective. Moreover,

g ker(i_N i_N(g) = 1_G/N

gN = N

g N.

Therefore, ker(i_N) = N.

The homomorphism i_N:GG/N is called the canonical homomorphism (associated with N). Note that by Proposition 6, every normal subgroup N of G appears as the kernel of a homomorphism of G, the canonical homomorphism.

Suppose that G, H, and J are groups, and suppose that a:GH, b:HJ, and c:GJ are homomorphisms. It is easy to check the composite function ba:GJ is also a homomorphism. We can picture this situation by drawing the following diagram:

We see from the diagram that there are two homomorphisms defined from G into J, ba and c. If c = ba, then we day that the diagram is commutative. In other words, if the diagram is commutative, then the functions defined by the two paths from G to J are the same.

Let f :GH be a homomorphism. Then we can construct the following diagram:

Here i is the canonical homomorphism i_ker(f). A common sort of question which arises in many contexts is: Does there exist a homomorphism j:G/ker(f)H [the dashed arrow in (5)] which makes the diagram commute? We will prove the existence and uniqueness of j in the following theorem.

Theorem 7 (1st [or fundamental] Isomorphism Theorem): Let f :GH be a homomorphism. Then there exists a homomorphism j :G/ker(f)H which makes diagram (5) commute. Such a j is unique. Moreover, j is an injection, so that j is an isomorphism and im(f) G/ker(f), where im(f) denotes the image of f. If, in addition, f is surjective, then j is surjective and H G/ker(f).

Remarks:

1. In many books, the first isomorphism theorem consists of the statement that if f :GH is a surjective homomorphism, then H G/ker(f). Indeed, this is how the first isomorphism theorem is usually applied. However, for many purposes it is important to know not only that H and G/ker(f) are isomorphic, but that there is an isomorphism which makes the diagram (5) commute, which makes them isomorphic.

2. The real utility of Theorem 7 (as well as the other isomorphism theorems) is that they allow one to conclude that two groups are isomorphic to one another without actually constructing the isomorphism. Thus, these isomorphism theorems allow one to build a calculus for computing with groups.

3. The fundamental theorem asserts that G/ker(f) can be identified with the image of f under the isomorphism of j. Moreover, once this identification is carried out, the commutativity of the diagram implies that the homomorphism f :GH is identified with the canonical homomorphism. Thus, roughly speaking, all homomorphisms f :GH are "the same as" canonical homomorphisms. This is the real impact of the fundamental theorem.

Example 6: Let f :G{1_G} be the homomorphism defined by f(g) = 1_G for all g G. Then f is surjective and ker(f) = G, and from the theorem we conclude that G/G {1_G}.

Example 7: Let f :D_n[F] be the homomorphism defined by

Then f is surjective and ker(f) = [R]. Then from the theorem we conclude that D_n/[R] [F], a fact which could be easily verified directly.

Proof of Theorem: Let K = ker(f). Then a typical element of G/K is of the form gK (g G). Since i(g) = gK, if (5) commutes, then we must have

Therefore, if j exists, it is unique and must be given by (6). Let us try to define j by (6). First, we must check that the definition of j(gK) depends only on the coset gK and not on the choice of g. But if gK = g'K, then gg'^-1 K, say gg'^-1 = k, so that f(gg'^-1) = 1_H, since K = ker(f). But then, since f is a homomorphism, 1_H = f(gg'^-1) = f(g)f(g')^-1, which implies that f(g) = f(g'). Therefore, the definition (6) makes sense. It is clear that j makes the diagram (5) commute. Moreover, j is a homomorphism since

Finally, j is an isomorphism, since if gK ker(f), then f(g) = 1_H, which implies that g K and thus gK = K. Therefore, ker(j) = {1_G/K}, and j is an isomorphism.