Let 
denote a multiplicative
group and 
a nonempty set.
An action
of on is described by a mapping
We abbreviate this by saying that acts
on or simply by calling
a - set or by writing
in short, since acts from the
left on .
Before
we provide examples, we mention a second but equivalent formulation.
A homomorphism 
from into the symmetric group
on is called a permutation
representation of on .
It is easy to check (exercise 
)
that the definition
of action given above is equivalent to
The kernel of will be
denoted by 
, and so we have, if 
,
the isomorphism
In the case when 
, the action is
said to be faithful.
A very trivial example is the natural action
of 
on
itself, where
the corresponding permutation representation
is the identity mapping. A number of less trivial examples will follow
in a moment.
Exercises
E 
.
Assume to be a -set and check carefully that
is in fact a permutation representation, i.e. that
and that 
.
