SemiLinearDecomposition( module, S, C, e )
module is a module for a matrix group G over a finite field GF(q).
The function returns true if G is found to act semilinearly.
G is assumed to act absolutely irreducibly. S is a set of invertible
matrices, generating a subgroup of G, and assumed to act irreducibly
but not absolutely irreducibly on the underlying vector space of
module. The matrix C centralises the action of S on the underlying
vector space and so acts as multiplication by a field generator x of
GF(q^e) for some embedding of a d/e-dimensional vector space over
GF(q^e) in the d-dimensional space. If C centralises the action of
the normal subgroup < S > ^G of G, then < S
> ^G embeds in GL(d/e,q^e), and G embeds in Gamma
L(d/e,q^e), the group of semilinear automorphisms of the
d/e-dimensional space. This is verified by the construction of a map
from G to Aut(GF(q^e)). If the construction is successful, the
function returns true. Otherwise a conjugate of an element of S is
found which does not commute with C. This conjugate is added to S
and the function returns false.
SemiLinearDecomposition is called by SmashGModule.
The algorithm is described in [6].
GAP 3.4.4