Subgroup
For a group G , a non-empty subset H ⊆ G is a subgroup of G , denoted by H ≤ 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 ∈ H we have that e H ⋅ h = h , but then e H ∈ H so that e H ⋅ e H = e H . On that same thought e H ∈ H ⊆ G so that e H ∈ G and thus e H ⋅ 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 ⟺ ( e H − 1 ) e H e H = ( e H − 1 ) e H e G ⟺ 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 S be the family of subgroups such that each one contains S , then it's defined as ⋂ 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 { x y x − 1 y − 1 : x , y ∈ G } as [ G , G ]