Dear GAP-Forum,
Dear Ashvin,

On Fri, 21 Jun 2002, ashvin db wrote:

--> I gave the following commands in GAP.
--> gap> z := ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16)
--> gap> h :=(1,4,6,7);
--> gap> test := h^-1 * z;
--> ( 1, 2, 8, 9,10,11,12,13,14,15,16)( 3, 4)( 5, 6)

Actually, I got a slightly different result from GAP at this point:

( 1, 8, 9,10,11,12,13,14,15,16)( 2, 3, 4)( 5, 6)

Anyway ...

--> assuming ( 1, 2, 8, 9,10,11,12,13,14,15,16)( 3, 4)( 5, 6)  are the k 
--> surviving subpaths after using the splitting template h for z.
--> what command do I use to get a k-cycle of all the heads of the surviving 
--> arcs.
--> in this case I have to get a 4 cycle (1,3,5,7).

... from your example it seems clear to me what you are interested in.
The desired 4-cycle consists of the "starting points" of the orbits on
the range [1..16] under the permutation 'test'. You can produce this
cycle by the following commands:

#-------------------- determine starting points:
sp := [];
for orbit in OrbitsPerms( [test], [1..16] ) do
   Add( sp, orbit[1] );
od;

#---------------------------------- build cycle:
k := Length( sp );
cyc_list := [1..sp[k]]; ## describes identity
for i in [1..k-1] do
   cyc_list[sp[i]] := sp[i+1]; 
od;
cyc_list[sp[k]] := sp[1];
## now 'cyc_list' describes the desired k-cycle
cyc := PermList( cyc_list );

Please note also that considering "Orbits( Group([test]) )" instead of
"OrbitsPerms( [test], [1..16] )" in the first for-loop would ignore
the fixed point 7 because orbits of a group <G> are orbits on the set
MovedPoints( <G> ) by default!

Best regards
Mathias

----------------------------------------------------------------------
Dr. Mathias Kratzer                   | I_nstitute for 
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