For all humankind
Academicsubsite
ZixuanZhang
ZixuanZhang
Ponder...

Basic Group Identities

In a group, right inverses are left inverses, left identities are right identities, inverses are unique, and the identity is unique.

Proposition 1.4

Let be a group, and .

  1. If , then . [right inverses are left inverses.]
  2. [left identities are right identities.]
  3. If , then . [inverses are unique.]
  4. If , then . [the identity is unique.]

Proof

For a right inverse of , choose a right inverse of and use associativity to obtain . This gives left identities as right identities. The same identities show that two inverses of are equal, and that two identity elements are equal.

Stated in