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
, we can state that for every element g
g@e = e@g = g
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. Loop
s, which include all groups, also require identity elements.
In the cyclic 4-group,
. | 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 numbers1, 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).
For multiplication to form a group with the real numbers (or any subset), we have to remember to exclude the number 0