Group (mathematics)

In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. For example, the set of integers is a group under the operation of addition. The branch of mathematics which studies groups is called group theory.

Simple theorems

• A group has exactly one identity element.
• Every element has exactly one inverse.
• You can perform division in groups; that is, given elements a and b of the group G, there is exactly one solution x in G to the equation x * a = b and exactly one solution y in G to the equation a * y = b.
• The expression "a₁ * a₂ * ··· * a_n" is unambiguous, because the result will be the same no matter where we place parentheses.
• (Socks and shoes) The inverse of a product is the product of the inverses in the opposite order: (a * b)⁻¹ = b⁻¹ * a⁻¹.

These and other basic facts that hold for all individual groups form the field of elementary group theory.

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