action.md

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Group actions

A group action of a group $G$ on a set $\Omega$ (from the right) is defined by a map $\mu:\Omega\times G\to \Omega$ that satisfies the compatibility conditions $\mu(\mu(x,g),h) = \mu(x, gh)$ and $\mu(x, 1_G) = x$ for all $x\in\Omega$.

The maps $\mu$ are implemented as functions that take two arguments, an element $x$ of $\Omega$ and a group element $g$, and return the image of $x$ under $g$.

In many cases, a natural action is given by the types of the elements in $\Omega$ and in $G$. For example permutation groups act on positive integers by just applying the permutations. In such situations, the function ^ can be used as action function, and ^ is taken as the default whenever no other function is prescribed.

However, the action is not always determined by the types of the involved objects. For example, permutations can act on vectors of positive integers by applying the permutations pointwise, or by permuting the entries; matrices can act on vectors by multiplying the vector with the matrix, or by multiplying the inverse of the matrix with the vector; and of course one can construct new custom actions in situations where default actions are already available.

Thus it is in general necessary to specify the action function explicitly, see the following sections.

Common actions of group elements

on_tuples
on_sets
permuted
on_indeterminates
on_lines
on_echelon_form_mats
on_subgroups

G-Sets

The idea behind G-sets is to have objects that encode the permutation action induced by a group (that need not be a permutation group) on a given set. A G-set provides an explicit bijection between the elements of the set and the corresponding set of positive integers on which the induced permutation group acts, see action_homomorphism(Omega::GSetByElements{T}) where T<:GAPGroup. Note that the explicit elements of a G-set Omega can be obtained using collect(Omega).

gset(G::Union{GAPGroup, FinGenAbGroup}, fun::Function, Omega)
permutation
acting_group(Omega::GSetByElements)
action_function(Omega::GSetByElements)
action_homomorphism(Omega::GSetByElements{T}) where T<:GAPGroup
is_conjugate(Omega::GSet, omega1, omega2)
is_conjugate_with_data(Omega::GSet, omega1, omega2)
orbit(Omega::GSetByElements{<:GAPGroup, S}, omega::S) where S
orbit(G::PermGroup, omega)
orbits(Omega::T) where T <: GSetByElements{TG} where TG <: GAPGroup
is_transitive(Omega::GSet)
is_regular(Omega::GSet)
is_semiregular(Omega::GSet)

Stabilizers

stabilizer(G::GAPGroup, pnt::Any, actfun::Function)
stabilizer(Omega::GSet)