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.