Sameed Zahoor
Joined: 12 Mar 2008, 10:57 Posts: 69 Location: India

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.( )

deant
Joined: 12 Jul 2010, 07:44 Posts: 154

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 (03 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.
