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March 30th, 2011, 05:06 PM   #1
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Isomorphic Groups..U(8) and Z4

Prove or disprove U( isomorphic to Z4

U( = {1,3,5,7) Z4= {0,1,2,3}
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March 30th, 2011, 06:04 PM   #2
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Re: Isomorphic Groups..U(8) and Z4

How many elements of order 2 are in ?
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March 31st, 2011, 05:40 AM   #3
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Re: Isomorphic Groups..U(8) and Z4

3 I believe. Does that make it non-isomorphic?
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March 31st, 2011, 07:23 AM   #4
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Re: Isomorphic Groups..U(8) and Z4

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Originally Posted by jcrot30
3 I believe. Does that make it non-isomorphic?
All four have order 2, since
1 = 1
9 = 1
25 = 1
49 = 1 (mod 8)

But this is not the case in Z4.

Have you considered Z2 x Z2 ??
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March 31st, 2011, 08:39 AM   #5
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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)
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March 31st, 2011, 08:51 AM   #6
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Re: Isomorphic Groups..U(8) and Z4

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Originally Posted by Turgul
...
(as a side note, technically you are correct jcrot30, that there are only 3 elements of order 2, since 1 is of order 1)
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