The Fundamental Group of Circle, Radius 1
Consider the space
Let
and let a be a loop in
with base point
which goes clockwise once around the circle. Let the set of loops inbe labeled
For each
and
define
where
is the identity function.
Note that
if and only if
For
there issuch that
Thus there is an isomorphism
between the quotient group
and 
where
if
and
loop the same number of times in the same sense, given by h([n,a]) =n.
The group
is generated by the element