| | | Generators of induced actions |
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.
| | | Generators of induced actions |