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
> 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
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