Isomorphic groups

**didact273** · March 22nd 2009, 12:06 AM

Prove that Aut(V) is isomorphic to $S_3$ and that $Aut(S_3)$ is isomorphic to $S_3$ . V is the four group.

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

I appreciate any help.

**SimonM** · March 22nd 2009, 03:53 AM

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)= $S_3$ .

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

$\theta (0) = 0$

Let $\theta(1) = c$

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

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

For this to be a bijection $c = \pm 1$ and we're done

**ThePerfectHacker** · March 22nd 2009, 09:47 AM

Originally Posted by didact273

Prove that Aut(V) is isomorphic to $S_3$ and that $Aut(S_3)$ is isomorphic to $S_3$ . V is the four group.

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

I appreciate any help.

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

**didact273** · March 22nd 2009, 02:25 PM

Originally Posted by ThePerfectHacker

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

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

Thanks.

**ThePerfectHacker** · March 22nd 2009, 07:07 PM

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 $S_3$ ? An example of one of them?

Thanks.

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

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