MatrixGroup() or order() incorrect for G_2(F_3)

[robharron]

If I use the generators for the exceptional group G_2(F_3) in its natural 7-dimensional representation over F_3 in the MatrixGroup? constructor, the order of the group returned is 8491392 (twice what it should be). However, if I use the same generators through the gap console within sage, I get the correct size.

K=GF(3)
sage: gens=[matrix(K,7,[
....: [0,2,0,0,0,0,0],
....: [2,0,0,0,0,0,0],
....: [0,0,0,2,0,0,0],
....: [0,0,2,0,0,0,0],
....: [0,0,0,0,0,2,0],
....: [0,0,0,0,2,0,0],
....: [2,2,2,2,1,1,1]]),
....: matrix(K,7,[
....: [2,0,0,0,0,0,0],
....: [0,0,2,0,0,0,0],
....: [1,1,1,0,0,0,0],
....: [0,0,0,0,2,0,0],
....: [1,0,0,1,1,0,0],
....: [0,0,0,0,0,0,2],
....: [1,0,0,0,0,1,1]])]
sage: M = MatrixGroup(gens); M.order()
8491392
sage: M.gap().Size()
8491392
sage: M.gap().Order()
8491392
sage: gap.Group([gap(gens[0]),gap(gens[1])]).Order()
8491392

gap> m1:= [
> [0,2,0,0,0,0,0],
> [2,0,0,0,0,0,0],
> [0,0,0,2,0,0,0],
> [0,0,2,0,0,0,0],
> [0,0,0,0,0,2,0],
> [0,0,0,0,2,0,0],
> [2,2,2,2,1,1,1]
> ]*Z(3);
gap> m2:= [
> [2,0,0,0,0,0,0],
> [0,0,2,0,0,0,0],
> [1,1,1,0,0,0,0],
> [0,0,0,0,2,0,0],
> [1,0,0,1,1,0,0],
> [0,0,0,0,0,0,2],
> [1,0,0,0,0,1,1]
> ]*Z(3);
gap> Order(Group( m1, m2 ));
4245696

[chapoton]

Compare

sage: L = [gap(GF(3)(i)) for i in range(3)]; L
[0*Z(3), Z(3)^0, Z(3)]

and

gap> [0, 1, 2]*Z(3);
[ 0*Z(3), Z(3), Z(3)^0 ]

[chapoton]

same problems for other finite fields of prime order

sage: p=5; L = [gap(GF(p)(i)) for i in range(p)]; L
[0*Z(5), Z(5)^0, Z(5), Z(5)^3, Z(5)^2]

versus

gap> p:=5; [0..p-1]*Z(p);
5
[ 0*Z(5), Z(5), Z(5)^2, Z(5)^0, Z(5)^3 ]

[chapoton]

This boils down to

gap> 1*Z(5);
Z(5)
gap> One(Z(5));
Z(5)^0

which is indeed not very-good looking.

[chapoton]

ok, so this must come from a confusion: in Gap, Z(5) is not the one of the finite field, but a generator of the group of invertible elements. This means that your matrices are not the correct ones, I think.

I propose to close this as invalid.

[tscrim]

Confirmed:

gap> One(GF(5));
Z(5)^0

and replacing by the above yields:

gap> Order(Group( m1, m2 ));
8491392

From the ​GAP manual:

The root returned by Z is a generator of the multiplicative group of the finite field with p^d elements, which is cyclic.

[embray]

Closing tickets in the sage-duplicate/invalid/wontfix module with positive_review (i.e. someone has confirmed they should be closed).