- #1
- 2,076
- 140
Homework Statement
Doing some suggested exercises out of my textbook today, there were two that I had trouble with.
1. Let G be a group and |G| = n. Suppose k is an integer relatively prime to n. Show that the mapping [itex]\phi : G → G \space | \space \phi(g) = g^k[/itex] is injective. If G is also abelian, show that the map [itex]\phi[/itex] is also an automorphism of G.
2. Suppose that [itex]\phi : \mathbb{Z_3} \oplus \mathbb{Z_5} → \mathbb{Z_{15}} \space | \space \phi((2,3)) = 2 \space[/itex] is an isomorphism. Find an element in [itex]\mathbb{Z_3} \oplus \mathbb{Z_5}[/itex] that maps to 1 in [itex]\mathbb{Z_{15}}[/itex].
Homework Equations
An automorphism is a bijective homomorphism.
An isomorphism is a bijective map.
The Attempt at a Solution
1. Okay, so G is a group and |G| = n. We also know gcd(n,k) = 1 for some integer k which means that gcd(n,k) = 1 = an + bk for some integers a and b. I'm pretty sure this group is under multiplication, but I could be wrong.
- We want to show the map is one to one. So we assume [itex]\phi(r) = \phi(s)[/itex] and we want to show r = s for any r and s in G.
[itex]\phi(r) = \phi(s)[/itex]
[itex]r^k = s^k[/itex]
Now here is where I get stuck, I'm pretty sure I have to use the fact that n and k are relatively prime here.
Ill save question 2 for after this one is finished. Any help is appreciated.