Compute group automorphisms using magma

automorphism-groupcomputer-algebra-systemsgroup-theorymagma-cas

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:

  1. Take for instance $f_1$. What do the $[0, 1]$ and $[2,2]$ mean under $A$ and $B$, respectively?
  2. Take again $f_1$. What does $f_1(A) \mapsto [1,1]$ mean? (e.g. $AB$?)
  3. 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.

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]

should be

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]

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 $G$...

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