Homomorphism?

Question

Why can't there be a homomorphism from Z_16 onto Z_2 cross Z_2? Please explain

Answer 1

It's very simple: Suppose φ:Z_16 → Z₂×Z₂ is a homomorphism. Let K = ker φ, then by the first isomorphism theorem, Z_16/K ≃ Z₂×Z₂. Therefore, we have 16/|K| = |Z_16|/|K| = |Z_16/K| = |Z₂×Z₂| = 4, so |K| = 16/4 = 4. Since Z_16 is cyclic, so is K, so K is generated by an element of order 4 in Z_16, and there are only two such elements, namely [4] and [12], both of which generate the same subgroup, namely 4Z_16. So K = 4Z_16, thus Z_16/4Z_16 ≃ Z₂×Z₂. But Z_16/4Z_16 ≃ Z₄, so by transitivity of isomorphism, we have Z₄ ≃ Z₂×Z₂. But this is impossible, since Z₄ contains an element of order 4, and Z₂×Z₂ does not. Therefore, no homomorphism can possibly exist from Z_16 to Z₂×Z₂. Q.E.D.

Edit: mathman, note that he specified that the homomorphism had to be onto. That means surjective.

To everyone else -- no, I'm not even going to comment.

Answer 2

Your question is not well formulated. Do you
look at Z_16 and Z_2 x Z_2 as abelian groups
or unital rings?

For rings, there is at least
the trivial homomorphism sending all of the elements
of Z_16 to the unit of Z_2 x Z_2.

Answer 3

Because abstract algebra studies sets with operations that generate interesting structure or properties on the set, the most interesting functions are those which preserve the operations. These functions are known as homomorphisms.

Consider the natural numbers with addition as the operation. A function which preserves addition should have this property: f(a + b) = f(a) + f(b).
For example, f(x) = 3x is one such homomorphism,
since f(a + b) = 3(a + b) = 3a + 3b = f(a) + f(b).
Note that this homomorphism maps the natural numbers back into themselves.

Answer 4

Be MORE SPECIFIC...what class is this from? Never heard the term homomorphism before when related to a math class!

Answer 5

Homomorphism is when you start acting really feminine isn't it?

Answer 6

You should probably ask someone in calculus class.

Answer 7

Im only in pre-algebra and i dont get tht

Answer 8

what are you talking about

Answer 9

is that where you morph into homo?

Answer 10

huh?