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Sunday, November 29, 2009

Extending C2 x C2 to Q8

What I never understood is that in books on Group
Theory 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 -> Q8
{ 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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