236 CHAPTER 4 Abstract Algebra
Here’s a much less obvious example. Consider the two infinite groups
(R,+) and (R+,·). At first blush these would seem quite different.
Yet, if we consider the mappingf :R → R+ given byf(x) =ex(the
exponential function) thenf is not only a bijection (having inverse ln)
but this mapping actually matches up the two binary operations:
f(x+y) = ex+y = ex·ey = f(x)·f(y).Notice that the inverse mappingg(x) = lnx, does the same, but in the
reverse order:
g(x·y) = ln(x·y) = lnx+ lny = g(x) +g(y).The point of the above is that through the mappingsfand its inverse
gwe see that group structure of (R+,·) is faithfully represented by the
group structure of (R+,·), i.e., the two groups are “isomorphic.” We
shall formalize this concept below.
Definition of Homomorphism: Let (G,∗) and (H,?) be groups, and
letf : G → H be a mapping. We say that f is a homomorphism
if for all g, g′ ∈ Gwe have f(g∗g′) = f(g)? f(g′). In other words,
in finding the image of the product of elements g, g′ ∈ G, it doesn’t
matter whether you first compute the product g∗g′ in G and then
applyf or to first apply f to gandg′and then compute the product
f(g)? f(g′) inH.
Of course, we now see that the exponential mapping from (R,+) to
(R+,·) is a homomorphism.
Here’s another example. Recall the group GL 2 (R) of 2×2 matrices
having real coefficients and non-zero determinants. Since we know that
det(A·B) = det(A)·det(B) we see that det : GL 2 (R) → R∗ is a
homomorphism, whereR∗denotes the multiplicative group of non-zero
real numbers.
Definition of Isomorphism: If f : G→ H is a homomorphism of
groups (G,∗) and (H,?), we say that f is an isomorphism if f is