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 Post subject: Is SU(1) isomorphic to U(1)?Posted: Thu, 12 Apr 2012 08:26:21 UTC
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Joined: Sun, 24 Jul 2011 23:27:16 UTC
Posts: 33
I've been told that SU(1) and U(1) are isomorphic as groups, is this really true? I know that
SU(1) would be all 1x1 matrices (i.e., numbers) with determinant = 1 and the determinant of
a 1x1 matrix is the entry itself (I think), while U(1) is the circle group. Hence, if my def. of the
determinant of a 1x1 is correct, |SU(1)| = 1 while |U(1)| is definitely not one element.

Last edited by jakey34 on Thu, 12 Apr 2012 08:45:43 UTC, edited 1 time in total.

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 Post subject: Re: Is SU(1) isomorphic to U(1)?Posted: Thu, 12 Apr 2012 08:40:00 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 7624
Location: NCTS/TPE, Taiwan
jakey34 wrote:
I've been told that SU(1) and U(1) are isomorphic as groups, is this really true? I know that
SU(2) would be all 1x1 matrices (i.e., numbers) with determinant = 1 and the determinant of
a 1x1 matrix is the entry itself (I think), while U(1) is the circle group. Hence, if my def. of the
determinant of a 1x1 is correct, |SU(1)| = 1 while |U(1)| is definitely not one element.

You mean "SU(1) would be all matrices with ...".

SU(1) is the trivial group. U(1) is the circle. They are not isomorphic.

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 Post subject: Re: Is SU(1) isomorphic to U(1)?Posted: Thu, 12 Apr 2012 08:46:29 UTC
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Joined: Sun, 24 Jul 2011 23:27:16 UTC
Posts: 33
Yes, I corrected it. Thanks for confirming my response!

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