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£SX and hÎH£SY 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