Properties of Homomorphism

  • Thread starter sleventh
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  • #1
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I am wondering what are the possible homomorphisms
[itex]\tau[/itex] : [itex]Z[/itex][itex]\overline{+}[/itex] -> [itex]Z[/itex][itex]\overline{+}[/itex]

From this it should be possible to determine which is injective, surjective, and which are isomorphic.

Homomorphisms between Z plus to Z plus will all be of the form [itex]\tau[/itex](x) = nx
since [itex]\tau[/itex](x) = [itex]\tau[/itex](1)[itex]\underline{1}[/itex] + [itex]\tau[/itex](1)[itex]\underline{2}[/itex] + ... + [itex]\tau[/itex](1)[itex]\underline{x}[/itex]

since we have a homomorphism and x is one summed x times.

all are injective

now im not sure how to tell which are surjective
 

Answers and Replies

  • #2
606
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I am wondering what are the possible homomorphisms
[itex]\tau[/itex] : [itex]Z[/itex][itex]\overline{+}[/itex] -> [itex]Z[/itex][itex]\overline{+}[/itex]

From this it should be possible to determine which is injective, surjective, and which are isomorphic.

Homomorphisms between Z plus to Z plus will all be of the form [itex]\tau[/itex](x) = nx
since [itex]\tau[/itex](x) = [itex]\tau[/itex](1)[itex]\underline{1}[/itex] + [itex]\tau[/itex](1)[itex]\underline{2}[/itex] + ... + [itex]\tau[/itex](1)[itex]\underline{x}[/itex]

since we have a homomorphism and x is one summed x times.

all are injective

now im not sure how to tell which are surjective

What kind of algebraic structure and under what operation(s) you think "Z plus" (the natural numbers, I presume?) is for you to talk about "homomorphisms"? Perhaps a monoid?
Tonio
 
  • #3
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What kind of algebraic structure and under what operation(s) you think "Z plus" (the natural numbers, I presume?) is for you to talk about "homomorphisms"? Perhaps a monoid?
Tonio
I'm guessing he means the group ##(\mathbb{Z},+)##, the group of integers under addition.

sleventh, you're right that each homomorphism can be written ##\tau_n(x) = nx## for an integer n. (Since ##\mathbb{Z}## is generated by 1, everything is determined by ##\tau_n(1)##.) Can you write out the range of the homomorphism for each n? There shouldn't be very many that are surjective.

Also, not all are injective. I can think of one homomorphism (a boring one, admittedly) that isn't.
 

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