> < ^ Date: Mon, 07 Feb 1994 08:53:00 +0100
^ From: Paul Igodt <paul.igodt@kulak.ac.be >
> ^ Subject: on permutation group "transversals".

We are initiating ourselves in the use of GAP. The following question arises
in looking at what GAP does to permutation groups.
It looks like if the man-page is not fully clear on this neither.

When one looks at the output of
gp.stabilizer
if gp is a well defined permutation group, than one finds a record containing
many "transversals". How should these transversals be understood? E.g. they contain
often many "extra comma's".

As an example consider the following output:

```Group( (1,2,3,4)(5,6,7,8), (1,2,3), (1,3,5), (1,2)(3,4,7) )
gp.transversal := [ (), (1,2)(3,4,7), (1,2,3), (1,2,3,4)(5,6,7,8), (1,3,5), (1,2,3,4)(5,6,7,8),
(1,2)(3,4,7), (1,2,3,4)(5,6,7,8) ]
gp.stabilizer :=
rec(
identity := (),
generators := [ (2,4,3,5,7)(6,8), (3,5,7), (2,6,7) ],
orbit := [ 2, 7, 5, 3, 4, 6, 8 ],
transversal :=
[ , (), (2,4,3,5,7)(6,8), (2,4,3,5,7)(6,8), (2,4,3,5,7)(6,8), (2,6,7),
(2,4,3,5,7)(6,8), (2,4,3,5,7)(6,8) ],
stabilizer := rec(
identity := (),
generators := [ (6,8), (3,5,7), (3,4,5), (5,8,7) ],
orbit := [ 3, 7, 5, 4, 8, 6 ],
transversal := [ ,, (), (3,4,5), (3,5,7), (6,8), (3,5,7), (5,8,7) ],
stabilizer := rec(
identity := (),
generators := [ (6,8), (4,5,7), (5,8,7) ],
orbit := [ 4, 7, 5, 8, 6 ],
transversal := [ ,,, (), (4,5,7), (6,8), (4,5,7), (5,8,7) ],
stabilizer := rec(
identity := (),
generators := [ (6,8), (5,8,7) ],
orbit := [ 5, 7, 8, 6 ],
transversal := [ ,,,, (), (6,8), (5,8,7), (5,8,7) ],
stabilizer := rec(
identity := (),
generators := [ (6,8), (6,7,8) ],
orbit := [ 6, 8, 7 ],
transversal := [ ,,,,, (), (6,7,8), (6,8) ],
stabilizer := rec(
identity := (),
generators := [ (7,8) ],
orbit := [ 7, 8 ],
transversal := [ ,,,,,, (), (7,8) ],
stabilizer := rec(
identity := (),
generators := [  ] ) ) ) ) ) ) )
```

Thanks a lot.

Is there a procedure in GAP to solve the "word"-problem for permutation groups?
E.g. something like "wordsolve(element, group, group.generators)" ?

Paul Igodt

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