- #1
sleventh
- 64
- 0
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 I am not sure how to tell which are surjective
[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 I am not sure how to tell which are surjective