MAT2006: Groups, Rings & Fields What is a group?

A group G is a set G together with an operation ∗, (that is, given a, b ∈ G there is an element a∗b ∈ G), satisfying the following conditions:

G1 (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c ∈ G (associativity).

G2 There is e ∈ G with e ∗ a = a = a ∗ e for all a ∈ G (identity).

G3 For each a ∈ G there is some b ∈ G with a ∗ b = e = b ∗ a (inverses).