# Isomorphic groups

• Mar 22nd 2009, 12:06 AM
didact273
Isomorphic groups
Prove that Aut(V) is isomorphic to $\displaystyle S_3$ and that $\displaystyle Aut(S_3)$ is isomorphic to $\displaystyle S_3$. V is the four group.

Also, prove that Aut(Z) is isomorphic to $\displaystyle I_2$. Z is the set of integers and $\displaystyle I_2$ is the integers mod 2.

I appreciate any help.
• Mar 22nd 2009, 03:53 AM
SimonM
1) Any automorphism on V will preserve order of each elements. Therefore it can permute i,j,k. There are no restrictions on the permutation. Hence Aut(V)= $\displaystyle S_3$.

2) Any isomorphism from the integers will have the form $\displaystyle \theta (a+b) = \theta (a)+\theta (b)$ (*)

$\displaystyle \theta (0) = 0$

Let $\displaystyle \theta(1) = c$

Then for all positive integers $\displaystyle \theta (n) = nc$ (induction using (*)

And for negative integers $\displaystyle \theta (-n) = -nc$

For this to be a bijection $\displaystyle c = \pm 1$ and we're done
• Mar 22nd 2009, 09:47 AM
ThePerfectHacker
Quote:

Originally Posted by didact273
Prove that Aut(V) is isomorphic to $\displaystyle S_3$ and that $\displaystyle Aut(S_3)$ is isomorphic to $\displaystyle S_3$. V is the four group.

Also, prove that Aut(Z) is isomorphic to $\displaystyle I_2$. Z is the set of integers and $\displaystyle I_2$ is the integers mod 2.

I appreciate any help.

Remember that $\displaystyle S_3 = \{ e,a,a^2,b,ab,a^2b\}$ where $\displaystyle a=(123),b=(12)$. Any automorphism on $\displaystyle S_3$ is determined by $\displaystyle \theta(a),\theta(b)$. The order of $\displaystyle \theta(a)$ is equal to order of $\displaystyle a$, thus, $\displaystyle \theta(a) = a,a^2$ similarly $\displaystyle \theta(b) = b,ab,a^2b$. In total we have at most $\displaystyle 6$ automorphism. Show that each one extens to an automorphism of $\displaystyle S_3$.
• Mar 22nd 2009, 02:25 PM
didact273
Quote:

Originally Posted by ThePerfectHacker
Remember that $\displaystyle S_3 = \{ e,a,a^2,b,ab,a^2b\}$ where $\displaystyle a=(123),b=(12)$. Any automorphism on $\displaystyle S_3$ is determined by $\displaystyle \theta(a),\theta(b)$. The order of $\displaystyle \theta(a)$ is equal to order of $\displaystyle a$, thus, $\displaystyle \theta(a) = a,a^2$ similarly $\displaystyle \theta(b) = b,ab,a^2b$. In total we have at most $\displaystyle 6$ automorphism. Show that each one extens to an automorphism of $\displaystyle S_3$.

I understand everything until the last part. Can you explain what you mean by each automorphism extending to an automorphism in $\displaystyle S_3$? An example of one of them?

Thanks.
• Mar 22nd 2009, 07:07 PM
ThePerfectHacker
Quote:

Originally Posted by didact273
I understand everything until the last part. Can you explain what you mean by each automorphism extending to an automorphism in $\displaystyle S_3$? An example of one of them?

Thanks.

I mean if $\displaystyle \theta(a) = a^2,\theta(b)=ab$. Then $\displaystyle \theta(a^2) = (a^2)^2=a$ and $\displaystyle \theta(ab) = \theta(a)\theta(b) = (a^2)(ab) = b$. So on and so forth. You will need to show that when you extend $\displaystyle \theta$, then $\displaystyle \theta$ would be an automorphism.