Actions of groups
Let G denote a multiplicative
group and X a nonempty set.
An action
of G on X is described by a mapping
G ´X -> X :(g,x) -> gx, such that
g(g'x) = (gg')x, and 1x=x.
We abbreviate this by saying that G acts
on X or simply by calling
X a G- set or by writing
GX,
in short, since G acts from the left on X.
Before
we provide examples, we mention a second but equivalent formulation.
A homomorphism d from G into the symmetric group
group
symmetric
SX:= { p | p:X -> X, bijectively }
on X is called a permutation
representation of G on X.
It is easy to check
that the definition of action given above is equivalent to
d:g -> bar (g), where bar (g) :x -> gx, is a permutation representation.
The kernel of d will be
denoted by GX, and so we have, if bar (G):= d[G],
the isomorphism
bar (G) simeq G/GX.
In the case when GX= {1 }, the action is
said to be faithful.
A very trivial example is the natural action
of SX on X itself, where
the corresponding permutation representation
d:p -> bar ( p)
is the identity mapping. A number of less trivial examples will follow
in a moment.
harald.fripertinger@kfunigraz.ac.at,
last changed: August 28, 2001
