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## Sunday, November 29, 2009

### Extending C2 x C2 to Q8

What I never understood is that in books on GroupTheory Q8 is shown as a concrete group, i.e. the group of quaternions {i, j, k | i^2 = j^2 = k^2 = -1, i*j=k, j*k=i, k*i=j } and not as an abstract group. Well, I just discovered that it is fairly easy to construct Q8 from C2 x C2 ( which is often shown in abstract form and concrete form: the Klein4 group ).The group C2 x C2 has the following presentation:<a,b | a^2 = b^2 = 1, a*b = b*a >.The group Q8-abstract has the following presentation:<a,b,c | a^2*c = b^2*c = 1, a*b*c = b*a >,members of this group are:{ 1, a, b, a*b, c, a*c, b*c, a*b*c }.The following isomorphism can be established between Q8-abstract and Q8:f: Q8-abstract -> Q8by{ 1 |-> 1,  a |-> i,   b |-> j,  ab |-> k,  c |-> -1  ac |-> -i,  bc |-> -j,  abc |-> -k }.

Q8 is not something like ( C2 X C2 ) : C2, where X stands for direct product and : stands for semi-direct product, but it is very likely something similar. I read briefly that there are ways to construct groups other than using the direct or semi-direct product. Will / must take some time to check this out.
The fact that Q8 can be constructed and has a fairly simple presentation predicts that there must be similar methods for -all- other finite groups.

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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)