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 \( G \)

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 \( [ G, G ] \)