As of May 4 2007 the scripts will autodetect your timezone settings. Nothing here has to be changed, but there are a few things

Please follow this blog

Search this blog

Tuesday, December 8, 2009

Aut( Q8 ) = S4

The automorphism group of the quaternion group is the symmetric group on four letters. Sounds fascinating to me. But once I figured it out it's rather easy.

The elements of Q8 are:
1:1, 2:-1, 3:i, 4:j, 5:k, 6:-i, 7:-j, 8:-k.
The generators are:
3:i, 4:j.

Then there are 24 isomorphisms, since there are 6 possible generators i, j, k, -i, -j, and -k but after 1 has been chosen, 4 remain ( choose i then -i can't be chosen anymore ).

So up to here I have shown that Aut(Q8)=24, not yet that it is isomorphic to S4. ( That's for later. )

No comments:

Post a Comment

Popular Posts

Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(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.)