Let G be a group and Aut G the set of all automorphisms of G. What is Aut(adsbygoogle = window.adsbygoogle || []).push({}); Z[itex]_{n}[/itex] for arbitrary n[itex]\in[/itex]N?

Sol

Let f[itex]\in[/itex]AutZ[itex]_{n}[/itex]. SinceZ[itex]_{n}[/itex]=<1>, f can be completely characterized by f(1), i.e., f(k) = f(1)[itex]^{m}[/itex], for k[itex]\in[/itex]Z[itex]_{n}[/itex] and m[itex]\in[/itex]N. By these facts, it follows that f(1) must generateZ[itex]_{n}[/itex], that is,Z[itex]_{n}[/itex]=<f(1)>.

Define H as the group of all generators ofZ[itex]_{n}[/itex] under multiplication. Then f(1)[itex]\in[/itex]H.

Now at this point I'm certain that AutZ[itex]_{n}[/itex][itex]\cong[/itex]H, but I just can't find a homomorphism from AutZ[itex]_{n}[/itex][itex]\rightarrow[/itex]H. I can't concoct an epimorphism in the opposite direction either. The closest I got is the below:

Also define the map c:AutZ[itex]_{n}\rightarrow[/itex]H by c(f) = f(1). Choose f,g[itex]\in[/itex]AutZ[itex]_{n}[/itex]. Then:

c(g[itex]\circ[/itex]f) = g[itex]\circ[/itex]f(1) = g(f(1)) = g(1)[itex]^{l}[/itex]. But any l'[itex]\equiv[/itex]l (mod n) [itex]\Rightarrow[/itex]g(f(1)) = g(1)[itex]^{l'}[/itex], so we can consider l' an equivalence class inZ[itex]_{n}[/itex]. That implies l = f(1)[itex]^{p}[/itex] for some p[itex]\in[/itex]N, so that c(g[itex]\circ[/itex]f) = f(1)[itex]^{p}[/itex]*g(1) = c(g)c(f)[itex]^{p}[/itex].

I can't see what I'm overlooking, so I'd appreciate if someone can help point me in the right direction. Thanks!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Help with classification of the set of automorphisms on Z/n

**Physics Forums | Science Articles, Homework Help, Discussion**