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## Tuesday, November 17, 2009

### Watched lecture 11 of Abstract Algebra E-222

Basically the same stuff as in lecture 10 with some cool examples. And finally the well known A = B A B^-1 matrix conjugation for base conversion.


GL(2,2) is the general linear group of dimension 2 over GF(2).
Gross asked which group is isomorphic to GL(2,2) ?
( I stopped the video and gave it a try. )

GF(2) has the following tables for addition and multiplication.
+    0 1    x    0 1
0    0 1    0    0 0
1    1 0    1    0 1

The possible maps from F2 -> F2 have the following matrices :
0 0    1 0    0 1    1 1
0 0    0 0    0 0     0 0

0 0    1 0    0 1    1 1
0 1    0 1    0 1    0 1

0 0    1 0    0 1    1 1
1 0    1 0    1 0    1 0

0 0    1 0    0 1    1 1
1 1    1 1    1 1     1 1

Elements of GL(2,2) are the matrices which have determinant 1.
1 0    1 1    0 1    1 1    1 0    0 1
0 1    0 1    1 0     1 0    1 1    1 1

Order    1    2    2    3    2    3

We now see that GL(2,2) is generated by
0 1    1 1
1 0     1 0
and is isomorphic to S3.

Gross however had a different ( smarter ) approach as follows.
F2 is the following set:
{ (0,0), (1,0), (0,1), (1,1) }
A linear transformation from F2 to F2 must fix (0,0)
so the elements of GL(2,2) are the permutations of
(1,0), (0,1) and (1,1) with group S3.



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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.)