(
Group Theory)
The definition of a
group requires that one element of each group, when combined with any other element of the group, yields that other element. If we symbolize the identity of a group
G by
e, we can state that for every element
g in
G,
g@e = e@g = g
(where
@ symbolizes the operation the group defines).
In addition, the identity element
e of a group G:
 is the only element of the group that is its own inverse:
e^{1}@e = e@e^{1} = e^{1} = e
 is in every subgroup of the group G
Please note: A
binary operation does not need to be a group in order to have an identity element.
Loops, which include all groups, also require identity elements.
Examples

In the cyclic 4group,
.  a b c d
+
a  a b c d
b  b c d a
c  c d a b
d  d a b c
a is the identity element.
 In the addition group on the set of real numbers, the identity
element is 0, since for each real number r,
0 + r = r + 0 = r
Since addition for integers (or the rational numbers, or any number of subsets of the real numbers) forms a normal subgroup of addition for real numbers, 0 is the identity element for those groups, too.
 In the multiplication group defined on the set of real numbers^{1}, the identity
element is 1, since for each real number r,
1 * r = r * 1 = r
And of course, this also holds for the rational numbers and many other subgroups (multiplication for integers is not a group).
^{1}For multiplication to form a group with the real numbers (or any subset), we have to remember to exclude the number
0.