Subgroups

Subgroup

For a group

G

, a non-empty subset

H \subseteq G

is a subgroup of

G

, denoted by

H \leq 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} ⟺ (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 \in G}

[G, G]