| | | **Generators of induced actions** |

#### Generators of induced actions

Given two permutation groups *G* and *H* we can compute the
direct sum or the
direct product of these two permutation groups.
Let *gÎG£S*_{X} and *hÎH£S*_{Y}
then you can compute elements
*f,yÎG´H* such that
the restriction of *f* onto *X* equals *g*,
the restriction of *f* onto *Y* is the identity,
the restriction of *y* onto *X* is the identity and
the restriction of *y* onto *Y* equals *h* for the direct sum,
and such that for the direct product

*f(x,y)=(gx,y) y(x,y)=(x,hy) *

hold, by using
INT dir_sum_perm(a,b,c,d) OP a,b,c,d;
INT dir_prod_perm(a,b,c,d) OP a,b,c,d;

In both cases `a b c`

and `d`

stand for the permutations
*g* *h* *g'* and *h'*.
Given the permutation groups *G* and *H* by systems of generators
you can compute the generators of the direct sum or the direct
product by

INT gen_dir_sum(a,b,c) OP a,b,c;
INT gen_dir_prod(a,b,c) OP a,b,c;

where `a`

and `b`

are the VECTORS of generators of
*G* and *H*. `c`

is the VECTOR of generators for the
corresponding permutation representation of the direct product
of the groups *G* and *H*.
The induced permutation representation of a PERMUTATION acting
on 2-sets can be computed with the (misnamed) procedure

INT m_perm_paareperm(a,b) OP a,b;

where `a`

is the given PERMUTATION and `b`

is the induced
PERMUTATION on the set of all pairs (for a certain labelling of the
pairs).
For a given set of generators you can compute a system of generators
of the induced action on the set of all 2-sets by

INT gen_on2sets(a,b) OP a,b;

where `a`

is a system of generators (PERMUTATION objects) and
`b`

is the system of the induced PERMUTATIONs on the set of all
2-sets.

harald.fripertinger@kfunigraz.ac.at,

last changed: November 19, 2001

| | | **Generators of induced actions** |