identity element of a group

The identity element of a group is its “do nothing” element: combine anything with it and that thing comes back unchanged. For addition the identity is 0 (n + 0 = n); for multiplication it is 1 (n · 1 = n); for the symmetries of a shape it is the move that leaves the shape exactly as it was. Every group is required to have one.

Formally, the identity is the element e satisfying e * a = a * e = a for every a in the group. The third group axiom demands its existence, and a short proof shows it must be unique: if two elements both behaved like identities, they would have to be equal. So a group has exactly one identity, no more and no less.

The identity is the anchor for the rest of the theory: inverses are defined relative to it (b is the inverse of a when a * b = e), and the order of an element is measured by how many times you must apply it to get back to e. Without a fixed reference point, neither idea would make sense.