(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 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).

1For multiplication to form a group with the real numbers (or any subset), we have to remember to exclude the number 0.

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