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November 18th, 2007, 02:00 PM   #1
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homomorphisms

Let D-sub-4 be the octic group and G'={e',a,b,c} be the Klein 4-group with e' the identify of G', and a^2=b^2=c^c=e'.

Tabulate a homomorphism Φ : D-sub-4 --> G' with the given requirements.

Φ((1)): e'
Φ((14)(23)): c
Φ((13)): a


We can tell easily that c^2=a^2=e' [But does that matter???]

But, I still need to figure out what maps to b.

The elements are:
(1), (1234), (13)(24), (1432), (24), (14)(23), (13), and (12)(34).

Again, we already know which ones map to e', a, and c. The elements that are left are:
(1234), (13)(24), (1432), (24), and (12)(34).
All of the elements squared are going to map to e' except (1234) and (1432).
This means that (13)(24), (24), and (12)(34) can be a, b, or c. But how do I know which is which?

Also, then what do (1234) and (1432) map to?
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