Let f: R -> R be the map on the reals R
defined by f(x) = -x.
- (a) Is f a group homomorphism of the reals, regarded as a group under addition?
If not, give an explicit example of some property of group homomorphisms that f does not have.
If f is a group homomorphism, determine whether or not it is an isomorphism, and explain
why it is, or is not, an isomorphism.
Answer: Yes, since f(x+y) = -(x+y) = -x + -y = f(x) + f(y).
- (b) Is f a ring homomorphism of the reals?
If not, give an explicit example of some property of ring homomorphisms that f does not have.
If f is a ring homomorphism, determine whether or not it is an isomorphism, and explain
why it is, or is not, an isomorphism.
Answer: No, since f(1) = -1 is not 1. Also, f(xy) = -xy is not f(x)f(y) = (-x)(-y) = xy.
- (c) Let h: Z -> (Z/5Z)x(Z/7Z) be the map defined
by h(m) = ([m]_{5},[m]_{7}). This map is in fact a homomorphism. Find an integer
m such that h(m) = ([3]_{5},[2]_{7}).
You may use any method to find such an m.
If you guess, however, you must verify that your m works. Otherwise, your work
should justify your answer.
Answer: We must solve the simultaneous equations
x == 3 (mod 5)
x == 2 (mod 7)
By the first equation, x = 3 + m5, so plugging into the second equation gives
3 + m5 = 2 + n7, or 7n - 5m = 3 - 2 = 1. We can either guess or use Euclid's algorithm
to find n = -2 and m = -3, so x = 3 + (-3)5 = -12. We use x = 2 + n7 to check:
x = 2 + 7(-2) = -12. Thus m = -12 works: h(-12) = ([3]_{5},[2]_{7}).