# Automorphisms and Order of a Map

1. Nov 24, 2008

### nobody56

1. The problem statement, all variables and given/known data
Let G={(a,b)/ a,b$$\in$$Z} be a group with addition defined by (a,b)+(c,d)=(a+c,b+d).
a) Show that the map$$\phi$$:G$$\rightarrow$$G defined by $$\phi$$((a,b))=(-b,a) is an automorphism of G.

b) Determine the order of $$\phi$$.

c) determine all (a,b)$$\in$$G with $$\phi$$((a,b))=(b,a)

2. Relevant equations

3. The attempt at a solution
a) multiple parts
1) Let (a,b)$$\in$$G, then $$\phi$$(a,b)=(-b.a), which is an element in G since -b,a$$\in$$Z therefore $$\phi$$(a,b)=(-b,a)$$\in$$G
2) One to One: suppose $$\phi$$(a,b)=$$\phi$$(c,d). That implies (-b,a)=(-d,c), which implies a=c and b=d therefore one to one.
3) Onto: Let (c,d)$$\in$$G to find (X,X)$$\in$$G such that $$\phi$$(X,X)=(c,d)=$$\phi$$((d,-c)) therefore onto
4) Operation Preserving: Let (a,b),(c,d)$$\in$$G. $$\phi$$((a+c,b+d))=(-(b+d),a+c) and $$\phi$$((a,b))+$$\phi$$((c,d))=(-b,a)+(-d,c)=(-b-d,a+c)=(-(b+d),a+c)
therefore $$\phi$$ is an isomophism, and since $$\phi$$ is from G to G, by definition $$\phi$$ is an automorphism of G

b) I am not sure where to start for the order of a map?

c) the set {(a,0)/a$$\in$$G} thus $$\phi$$((a,0))=(-0,a)=(0,a)

2. Nov 24, 2008

### sutupidmath

for part (ii) the order of $$\phi$$ is 5. (at least from my world over here, it looks like that).
I assume that the order of a mapping would be, whenever:

$$\phi^k=\phi$$ so looking at your problem

$$\phi^5(a,b)=\phi(a,b)=(-b,a)$$

Check the calculations, i did it really fast....

3. Nov 24, 2008

### nobody56

are the other two parts ok?

and you said 5 because....
$$\phi$$((a,b))=(-b,a)
$$\phi$$((-b,a))=(-a,-b)
$$\phi$$((-a,-b))=(b,-a)
$$\phi$$((b,-a))=(a,b)
$$\phi$$((a,b))=(-b,a)
right?
(I can not find anything to do with order of maps in my book)

4. Nov 24, 2008

### HallsofIvy

Staff Emeritus
Yes, the first two parts are ok.

The "order of a map" is, as sutupidmath said, the smallest positive integer, k, such that $\phi^k= \phi$. Did you look up "order" in the index of your book?

5. Nov 24, 2008

### nobody56

Yea, but I wasnt sure if I could apply the same rules to a map. In my book it just has for order is of a group or the order on an element.

Thanks for your help and for clarifing that.