A group action of a group 
on a set 
is a map 
such that the
following conditions hold for all elements 
.
1. 
where 
is the identity element
of 
.
2. 
for
all 
.
In this case, 
is called a transformation group,
is called a 
-set, and 
is called the group action.
In a group action, a group permutes the elements of 
. The identity element
does nothing, while a composition of group actions
corresponds to the action of the composition. For
example, as illustrated above, the symmetric group 
acts on the digits 0 to 9 by permutations.
For a given 
, the set 
, where the group action moves 
, is called the group orbit
of 
. The subgroup which fixes
is the isotropy
group of 
.
For example, the group ![Z_2={[0],[1]}](/images/equations/GroupAction/Inline22.svg)
acts on the real numbers
by multiplication by 
. The identity
element leaves everything fixed, while ![[1]](/images/equations/GroupAction/Inline24.svg)
sends 
to 
. Note that ![[1]·[1]=[0]](/images/equations/GroupAction/Inline27.svg)
, which corresponds to 
. For 
, the group
orbit of 
is 
, and the isotropy
group is the trivial group, ![{[0]}](/images/equations/GroupAction/Inline32.svg)
. The only group
fixed point of this action is 
.
In a group representation, a group acts by invertible linear
transformations of a vector space 
. In fact, a group
representation is a group homomorphism
from 
to 
, the general
linear group of 
. Some groups
are described in a group representation,
such as the special linear group, although
they may have different group representations.
An important example is the group action of the Galois group on the roots of a polynomial. However, there are numerous examples and applications of group actions in many branches of mathematics, including algebra, topology, geometry, number theory, and analysis, as well as the sciences, including chemistry and physics.
