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

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

I appreciate any help.

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- Mar 22nd 2009, 12:06 AMdidact273Isomorphic groups
Prove that Aut(

**V**) is isomorphic to and that is isomorphic to .**V**is the four group.

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

I appreciate any help. - Mar 22nd 2009, 03:53 AMSimonM
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**)= .

2) Any isomorphism from the integers will have the form (*)

Let

Then for all positive integers (induction using (*)

And for negative integers

For this to be a bijection and we're done - Mar 22nd 2009, 09:47 AMThePerfectHacker
- Mar 22nd 2009, 02:25 PMdidact273
- Mar 22nd 2009, 07:07 PMThePerfectHacker