Subgroups

Subgroups

判定子群

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 $a^{-1}$∈ H。
  or
• 对任意 a, b ∈ H, $a{b}^{-1}$∈ H

Definition

• 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 $H\ne G$ ($H\subset G$)

Properties

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 $h^{-1}\in H$.