Group Actions
Letbe a set and let
be a group who elements act on the set
A left group action is a function
such that:
for all
where
is the identity element in
for all
and
Right actions are similarly defined.
From these two axioms, it follows that for everythe function which maps
to
is a bijective map from
to
Therefore, one may alternatively and equivalently define a group action of
on
as a group homomorphism from
to the set of all bijective maps from
to
If a group actionis given, we also say that
acts on the set
or
is a
- set.
Every group
acts on itself in two natural ways:
for all
or
for all
The symmetric group
and its subgroups act on the set
by permuting its elements.
The symmetry group of a polyhedron acts on the set of vertices of that polyhedron.
The symmetry group of any geometrical object acts on the set of points of that object.
The automorphism group of a vector space (or graph, or group, or ring...) acts on the vector space (or set of vertices of the graph, or group, or ring...).
The Matrix groups
and
act on
The Galois group and every subgroup of a field extension
acts on the bigger field
The additive group of the real numbers
acts on the phase space of systems in classical mechanics (and in more general dynamical systems): if
and
is in the phase space, then
describes a state of the system, and
is defined to be the state of the system
seconds later if
is positive or
seconds ago if
is negative.