Subgroup

For a group $G$, a non-empty subset $H\u0e42\x8a\x86G$ is a subgroup of $G$, denoted by $H\u0e42\x89\u0e04G$, if and only if $H$ is also a group under the operation of $G$

The Identity in a Subgroup is the Same

Let $H$ be a subgroup of $G$, and let ${e}_{H},{e}_{G}$ be their identities respectively, then $$e}_{H}={e}_{G$$

Subgroup Generated by a Set

Let $S$ be a subset of a group $G$, then we define **the subgroup generated by $S$** the intersection of all subgroups that contain $S$. Symbolically let $\mathcal{S}$ be the family of subgroups such that each one contains $S$, then it's defined as $$\u0e42\x8b\x82\mathcal{S}$$

The Subgroup generated by S is the Smallest Subgroup containing S

Subgroup Generated by $xy{x}^{\u0e42\x88\x921}{y}^{\u0e42\x88\x921}$

Let $G$ be a group, then we denote the subgroup of $G$ generated by the set $\{xy{x}^{\u0e42\x88\x921}{y}^{\u0e42\x88\x921}:x,y\u0e42\x88\x88G\}$ as $[G,G]$