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Archived: 07 Aug 2014, 09:58
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Sameed Zahoor
Joined: 12 Mar 2008, 10:57 Posts: 69 Location: India
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 Exercise [13.04]
We are given, 1.  2.  3.  a)  We have,  (using 3) b)  Consider the identity  Now,  (using 3) or  or  (using 3 again) or  c)  We have,    (using 3)  (using 3)  d)  It is merely a restatement of 2.(  )
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12 Jul 2008, 10:38 |
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deant
Joined: 12 Jul 2010, 07:44 Posts: 154
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 Re: Exercise [13.04]
Given only 1, 2, and 3, there is no mention of any "-1" element. There are only 1, i, C, and various multiplications of these.
Now rule (1) allows you to cancel any 4 i's multiplied in a row; Rule (2) allows you to cancel any 2 C's multiplied in a row; and rule (3) allows you to take any 'string' of i's and C's, and move all the C's to the right, e.g.
CiiCi
= CiiiiiC (3) = CiC (1) = iiiCC (3) = iii (2)
So, each element of the group can be represented as (0-3 i's)(0 or 1 C). The 8 distinct elements are thus: 1, i, ii, iii, C, iC, iiC, and iiiC.
..of course there's no reason you can't then denote i^2 as '-1', but it makes no difference to the structure of the group.
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18 Jul 2010, 16:11 |
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