A group action is a triple (G,Omega ,m), where G is a group, Omega a set and m:Omega ×G®Omega a function (shose action is compatible with the group arithmetic). We call Omega the domain of the action.
In GAP, Omega can be a duplicate-free collection (an object that permits access to its elements via the Omega[n] operation, for example a list), it does not need to be sorted (see IsSet).
The acting function m is a GAP function of the form
actfun(pnt,g)
that returns the image m(pnt ,g ) for a point pnt Î Omega and a group element g Î G .
Groups always acts from the right, that is m(m(pnt ,g ),h ) = m(pnt ,gh ).
GAP does not test whether an acting function actfun satisfies the
conditions for a group operation but silently assumes that is does.
(If it does not, results are unpredictable.)
The first section of this chapter , About Group Actions, describes the various ways how operations for group actions can be called.
Functions for several commonly used action are already built into GAP. These are listed in section Basic Actions.
The following sections then describe operations to compute orbits, stabilizers, the induced permutation images of group actions as well as properties of actions.
Finally section External Sets describes the concept of ``external sets'' which represent the concept of a `` G-set'' and underly the actions mechanism.
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GAP 4 manual