Before we generalize this method of complementation we should recall 
.
It shows that each action of the form
can be considered as an action of 
on 
or
as an action of 
on 
. Hence each such action of
on 
gives rise to an action of 
on
and leads us to the discussion of the Involution
Principle.
We call a group element 
an involution
if and only if 
. The Involution Principle is a method of counting objects by simply
defining a nice involution 
on a suitably chosen set 
and using
the fact that 
has orbits of length 1 (the
selfenantiomeric orbits) and of length 2 (which form the enantiomeric pairs)
only. A typical example is the complementation of graphs
which is, for 
, an involution on the set of graphs on 
vertices. An even easier case
is described in
.
Examples We wish to
prove that the number of divisors
of 
is odd if and only if 
is a square. In order to do
this we consider the set 
of
these divisors and define
This mapping
is an involution, if 
, and obviously 
is odd if and only if
has a fixed point, i.e. if and only if
there exists a divisor 
such that 
, or,
in other words, if and only if 
.
A less trivial example is the following proof (due to D. Zagier) of the fact that every prime number which is congruent 1 modulo 4 can be expressed as a sum of two squares of positive natural numbers. Consider the set
The following map is an involution on 
(exercise ):
This involution has exactly one fixed point, namely 
, if 
,
therefore 
must be odd, and consequently the involution
possesses a fixed point, too, which shows that 
, a sum of two
squares.
Exercises
E 
.
Check the details of the second example in .