Thread: Example of a group action

1. Example of a group action

I'm trying to generate examples for group action on sets in the lines of first 2 chapters of Dummit & Foote. And maybe I'm doing something wrong.

I tried to define group action like this:

Take all 2x2 matrices with entries from Z2 and remove the ones with det=1. We've got 10 entries for the set. Let's call the set A.
The remaining (2^2-1)(2^2-2)=6 2x2 matrices (the ones with det=1) form a group - GL(2,2).

We then define a group action that maps GL(2,2) x A -> A by using plain matrix multiplication. Multiplication must work, since detAB=detA x detB.
So if You multiply a matrix from GL(2,2) by matrix with det=0, You get matrix with det=0. Moreover:

| 1 0 |
| 0 1 | a = a for any a in A and

g2 (g1 a) = (g2 g1) a for any g1, g2 in GL(2,2) and any a in A (this is simply the associativity of matrix multiplication).

So, if we fix particular g in G, we get function that maps A to A bijectively - a permutation.

Then the problem:

| 0 0 | | 0 0 | | 0 0 |
| 1 0 | | 0 0 | = | 0 0 |

and

| 0 0 | | 0 0 | | 0 0 |
| 1 0 | | 1 1 | = | 0 0 |

I'm starting to think I'm not multiplying right, or maybe somehow detAB=detA*detB does not work for matrices with elements from Z2
(have little experience with matrices with elements not from R).

Aghm, help???

2. Re: Example of a group action

What is the problem? You are multiplying two elements of $A$ together. You are not multiplying anything from $G$.

Oh, boy ....