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 \( [ G, G ] \)