We present here the functionality which allows to compute the Sylow subgroups of finite groups of Lie type.
This procedure prints out a list of all primes p dividing the order of the group of Lie type G along with the "goodness" of p, the exponent of p in the factorisation of |G| and a sequence of integers. The positive integers give the orders of the decomposition of a torus Tw into cyclic groups such that the Sylow subgroup is contained in < Tw, CW(w) >. The negative number indicates the p-part coming from CW(w). If more than one such torus exists, then one line is printed for each of them.A prime is said to be "GOOD" if it is equal to the characteristic of the base field k of G, "good" if the Sylow subgroup is abelian, thus contained in a torus, and "bad" if it is not abelian and thus not contained in a torus. See [Hal05] for the algorithm used.
Compute a p-Sylow subgroup S of the group of Lie type G. Returned is a list of a two sequences. The second sequence contains generators of S. The first one is a sequence of integers giving the orders of the respective generator if the generator is a torus element and the negative of the order of < g >/(< g > ∩Tw) in case the generator g is not a torus element. See [Hal05] for the algorithm used.
> G := GroupOfLieType("G2", 5);
> PrintSylowSubgroupStructure(G);
G: Group of Lie type G2 over Finite field of size 5
Order(G) is 2^6 * 3^3 * 5^6 * 7^1 * 31^1
Order(W) is 2^2 * 3^1
...compute tori...
...compute sylows...
2 (bad) : 6 [ 4, 4, -4 ]
3 (bad) : 3 [ 6, 6, -3 ]
5 (GOOD) : The unipotent subgroup of G
7 (good) : 1 [ 21 ]
31 (good) : 1 [ 31 ]
> SylowSubgroup(G,2);
[*
[ 4, 4, -2, -2 ],
[ (2 1) , (1 2) , n2 , n1 n2 n1 n2 n1 n2 ]
*]
note that the orders of the non-toral elements is not necessarily the corresponding integer
in the first sequence:
> gens := $1[2]; > [ Order(g) : g in gens ]; [ 4, 4, 4, 4 ]but, in this example, their squares are contained in the torus:
> gens[3]^2 eq gens[2]^2, gens[4]^2 eq gens[2]^2; true true