群 G 的非空子集 H 是 G 的子群,当且仅当 H 同时满足以下条件:

  • 若 a, b ∈ H,则 ab ∈ H;
  • 若 a ∈ H,则 ∈ H。
  • 对任意 a, b ∈ H, ∈ H


  • A subgroup H of a group G
    We define a subgroup H of a group G to be a subset H of G such that when the group operation of G is restricted to H, H is a group in its own right.

  • The trivial subgroup
    The subgroup H = {e} of a group G is called the trivial subgroup.

  • Proper subgroup
    H is a subgroup of G and ()


A subset H of G is a subgroup if and only if it satisfies the following conditions

  1. The identity e of G is in H.
  2. If h1, h2 ∈ H, then h1, h2∈ H.
  3. If h ∈ H, then .