```
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.
```

2-2018 Teaching by misleading

2 months ago

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