Subgroup
For a group $$G$$, a non-empty subset $$H \subseteq G$$ is a subgroup of $$G$$, denoted by $$H \le G$$, 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 $e _ H = e _ G$
Since $$e _ H$$ is the identity in $$H$$ that means for any other $$h \in H$$ we have that $$e _ H \cdot h = h$$, but then $$e _ H \in H$$ so that $$e _ H \cdot e _ H = e _ H$$. On that same thought $$e _ H \in H \subseteq G$$ so that $$e _ H \in G$$ and thus $$e _ H \cdot e _ G = e _ G$$ were we've used the fact that $$e _ G$$ is the identity in $$G$$. So we have the combined equality $e _ H e _ H = e _ H e _ G \iff \left( e _ H ^ { -1 } \right) e _ H e _ H = \left( e _ H ^ { -1 } \right) e _ H e _ G \iff e _ H = e _ G$ So, in fact their identities are the same.
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 $\bigcap \mathcal{ S }$
The Subgroup generated by S is the Smallest Subgroup containing S
Subgroup Generated by $$x y x ^ { -1 } y ^ { -1 }$$
Let $$G$$ be a group, then we denote the subgroup of $$G$$ generated by the set $$\left\{ x y x ^ { -1 } y ^ { -1 } : x, y \in G \right\}$$ as $$[ G, G ]$$