I'm learning how to use MAGMA to compute automorphism groups, and I have difficulty interpreting the output. Concrete (and functional) example:
G := SL(2,3);
print "generators of SL(2,3): ", Generators(G);
AG := AutomorphismGroup(G);
print "Aut( SL(2,3) ): ", AG;
Output (executed via the online MAGMA calculator):
generators of SL(2,3): {
[1 1]
[0 1],
[0 1]
[2 0]
}
Aut( SL(2,3) ): A group of automorphisms of SL(2, GF(3))
Generators:
Automorphism of SL(2, GF(3)) which maps:
[1 1]
[0 1] |--> [1 1]
[0 1]
[0 1]
[2 0] |--> [1 2]
[2 2]
Automorphism of SL(2, GF(3)) which maps:
[1 1]
[0 1] |--> [0 1]
[2 2]
[0 1]
[2 0] |--> [0 2]
[1 0]
Automorphism of SL(2, GF(3)) which maps:
[1 1]
[0 1] |--> [1 0]
[2 1]
[0 1]
[2 0] |--> [0 1]
[2 0]
Automorphism of SL(2, GF(3)) which maps:
[1 1]
[0 1] |--> [1 1]
[0 1]
[0 1]
[2 0] |--> [0 1]
[2 0]
Automorphism of SL(2, GF(3)) which maps:
[1 1]
[0 1] |--> [1 2]
[0 1]
[0 1]
[2 0] |--> [2 1]
[1 1]
So $SL(2,3)$ has two generators, the matrices $A := \begin{bmatrix}1&1 \\\ 0 & 1\end{bmatrix}$ and $B := \begin{bmatrix}0 &1 \\\ 2 & 0\end{bmatrix}$, and MAGMA says that $Aut(SL(2,3))$ is generated by four automorphisms $f_1, \ldots, f_4$ and specify how each $f_i$ acts on $A$ and $B$. What I don't understand about the MAGMA output:
- Take for instance $f_1$. What do the $[0, 1]$ and $[2,2]$ mean under $A$ and $B$, respectively?
- Take again $f_1$. What does $f_1(A) \mapsto [1,1]$ mean? (e.g. $AB$?)
- I know that MAGMA uses right group action. Does this convention extend to group automorphisms as well? In order words, does $f_i(xy) = f_i(x) f_i(y)$ or $f_i(y) f_i(x)$?
Thanks for your help!
Best Answer
To answer 1. and 2. together: I think there's just a whitespace error, i.e.
should be
As for 3, I don't think $f_i(xy) = f_i(y)f_i(x)$ makes much sense? Take $f_i = \operatorname{id}_G$ for any non-abelian group $G$...