ΘρϵηΠατπ

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 eH,eG be their identities respectively, then eH=eG
Since eH is the identity in H that means for any other h∈H we have that eH⋅h=h, but then eH∈H so that eH⋅eH=eH. On that same thought eH∈H⊆G so that eH∈G and thus eH⋅eG=eG were we've used the fact that eG is the identity in G. So we have the combined equality eHeH=eHeG⟺(eH−1)eHeH=(eH−1)eHeG⟺eH=eG 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 xyx−1y−1
Let G be a group, then we denote the subgroup of G generated by the set {xyx−1y−1:x,y∈G} as [G,G]