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March 30th, 2011, 05:06 PM  #1 
Newbie Joined: Mar 2011 Posts: 6 Thanks: 1  Isomorphic Groups..U(8) and Z4
Prove or disprove U( isomorphic to Z4 U( = {1,3,5,7) Z4= {0,1,2,3} 
March 30th, 2011, 06:04 PM  #2 
Senior Member Joined: Aug 2010 Posts: 195 Thanks: 5  Re: Isomorphic Groups..U(8) and Z4
How many elements of order 2 are in ?

March 31st, 2011, 05:40 AM  #3 
Newbie Joined: Mar 2011 Posts: 6 Thanks: 1  Re: Isomorphic Groups..U(8) and Z4
3 I believe. Does that make it nonisomorphic?

March 31st, 2011, 07:23 AM  #4  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Isomorphic Groups..U(8) and Z4 Quote:
1 = 1 9 = 1 25 = 1 49 = 1 (mod 8) But this is not the case in Z4. Have you considered Z2 x Z2 ??  
March 31st, 2011, 08:39 AM  #5 
Senior Member Joined: Aug 2010 Posts: 195 Thanks: 5  Re: Isomorphic Groups..U(8) and Z4
As The Chaz points out, every element of has the property that (the identity). Isomorphisms send the identity to the identity and respect the operation, so if were an isomorphism, we would have: and so for all . But since would be surjective, then every element would satisfy . Is this true? (as a side note, technically you are correct jcrot30, that there are only 3 elements of order 2, since 1 is of order 1) 
March 31st, 2011, 08:51 AM  #6  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Isomorphic Groups..U(8) and Z4 Quote:
 

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