> < ^ Date: Mon, 24 Jun 2002 10:26:30 +0200 (MESZ)
< ^ From: Mathias Kratzer <kratzer@exp-math.uni-essen.de >
< ^ Subject: Re: k-cycle of subpath heads

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
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 );
```

"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
E-Mail: kratzer@exp-math.uni-essen.de | E_xperimental
Phone : +49-201-183-7680              | M_athematics     Ellernstr. 29
Visit : IEM, Room 205                 |                  D-45326 ESSEN
```

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