(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.