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5 After successful recognition

This chapter explains, what one can do with recognition nodes after a successful recognition (and possibly verification).

Of course, one can inspect the whole tree of recognition nodes just by looking at the stored attribute values. Moreover, constructive membership tests can be performed using the function SLPforElement (3.2-15), thereby writing an arbitrary element in terms of the nice generators, which are stored in the attribute NiceGens (3.2-8). If fail is returned, then the element in question does not lie in the recognised group or the recognition made an error.

Here is an example of a successful recognition tree:

gap> g := DirectProduct(SymmetricGroup(12),SymmetricGroup(5));
Group([ (1,2,3,4,5,6,7,8,9,10,11,12), (1,2), (13,14,15,16,17), (13,14) ])
gap> ri := RecogniseGroup(g);
#I  Finished rank 90 method "NonTransitive": success.
#I  Going to the image (depth=0, try=1).
#I  Finished rank 95 method "MovesOnlySmallPoints": success.
#I  Back from image (depth=0).
#I  Calculating preimages of nice generators.
#I  Creating 20 random generators for kernel.
....................
#I  Going to the kernel (depth=0).
#I  Finished rank 80 method "Giant": success.
#I  Back from kernel (depth=0).
<recognition node NonTransitive
 F:<recognition node MovesOnlySmallPoints Size=120>
 K:<recognition node Giant Size=479001600>>

One sees that the recursive process runs, first it finds that the permutation action is not transitive, a homomorphism is found by mapping onto the action on one of the orbits. The image is recognised to permute only a few points. The kernel is recognised to be a full symmetric group in its natural action on at least 10 points (recognised as Giant).

After this, we can write arbitrary group elements in the group g in terms of the nice generators:

gap> x := PseudoRandom(g);
(1,12)(2,5,9,11,10,3,4)(7,8)(13,14,16,15,17)
gap> slp := SLPforElement(ri,x);
<straight line program>
gap> ResultOfStraightLineProgram(slp,NiceGens(ri));
(1,12)(2,5,9,11,10,3,4)(7,8)(13,14,16,15,17)

Note that this example only works by using also the recog package which contains the necessary recognition methods.

5.1 Functions and methods for recognition nodes

If you need an element explicitly written in terms of the original generators, you can use the following function:

5.1-1 SLPforNiceGens

Returns: an SLP expressing the nice generators in the original ones

This function assembles a possibly quite large straight line program expressing the nice generators in terms of the original ones by using the locally stored information in the recognition tree recursively.

You can concatenate straight line programs in the nice generators with the result of this function to explicitly write an element in terms of the original generators.

5.1-2 \in

Returns: true or false

This method tests, whether the element x lies in the group recognised by the recognition node ri. Note that this is only a convenience method, in fact SLPforElement (3.2-15) is used and the resulting straight line program is thrown away.

5.1-3 Size

Returns: the size of the recognised group

This method calculates the size of the recognised group by multiplying the size of the image and the kernel recursively. It is assumed that leaf nodes know already or can calculate the size of their group.

5.1-4 DisplayCompositionFactors

Returns: nothing

This function displays a composition series by using the recursive recognition tree. It only works, if the usual operation CompositionSeries (Reference: CompositionSeries) works for all leaves. THIS DOES CURRENTLY NOT WORK FOR PROJECTIVE GROUPS AND THUS FOR MATRIX GROUPS!

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