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

A subgroup of a group G is a subset H of G which is also a group under the same operation. In detail, H is a subgroup of G if H is a subset of G and

SG1 a ∗ b ∈ H whenever a ∈ H and b ∈ H (closure).

SG2 If e denotes the identity element of G, then e ∈ H (identity).

SG3 For each a ∈ H then a^{−1} ∈ H where a^{−1} denotes the inverse of a in G (inverses).

We write H ≤ G as shorthand for “H is a subgroup of G”.