Finding Elements with Prescribed Properties

RandomElementOfOrder(G, n : parameters): GrpMat, RngIntElt -> BoolElt, GrpMatElt, GrpSLPElt, BoolElt

Central   : BoolElt                        Default: false
Proof     : BoolElt                        Default: true
Randomiser: GrpRandProc                    Default: 
MaxTries  : RngIntElt                      Default: 100

Given a finite matrix group \(G\), this intrinsic attempts to locate an element \(x\) of order \(n\) in \(G\) by random search. If such an element is found, then the return values are the boolean value true, the element \(x\), and an SLP for this element.

If Central is true, then an element is sought which has order \(n\) modulo the centre of \(G\). If Proof is false, then the element returned may have order a multiple of \(n\). In either case, the final return value indicates whether the element returned is known to have the precise order. The parameter MaxTries specifies the maximum number of random elements that are chosen. The parameter Randomiser specifies the random process that is to be used to construct the element and the SLP returned for the element is in the word group associated with this process. The default value of Randomiser is the process RandomProcessWithWords(G).

RandomElementOfNormalClosure(G, N): Grp -> GrpElt

Given a group \(G\) and a subgroup \(N\) of \(G\), this intrinsic returns a random element of the normal closure of \(N\) in \(G\). Note that \(G\) may be a permutation or matrix group. The algorithm is due to Leedham-Green and O’Brien [Leedham-Green and O'Brien, 2002].

InvolutionClassicalGroupEven(G : parameters): GrpMat[FldFin] -> GrpMatElt[FldFin], GrpSLPElt, RngIntElt

SmallCorank: BoolElt                      Default: false
Case       : MonStgElt                    Default: "unknown"

Let \(G\) be a quasisimple classical group in its natural representation and in even characteristic. If \(G\) is of type \(\Omega^+\) or \(\Omega^-\) then it must have even degree at least \(4\) and be defined over a field with at least \(4\) elements. The corank of an involution \(I\) is the rank of \(I-\)Identity\((G)\). This function returns an involution \(I\) of corank in \([d/4, \ldots, d/2]\), the SLP for \(I\) in WordGroup\((G)\), and the corank of the involution. The parameter Case should be one of "SL", "Sp", "SU", "Omega-", or "Omega+". If SmallCorank is true, then accept involution of small corank. The algorithm used to construct the involution is described in [Dietrich et al., 2013]; it was implemented by Heiko Dietrich.