# Math Help - Abstract Algebra II: Normal Subgroups

1. ## Abstract Algebra II: Normal Subgroups

Let H be a subgroup of a group G. Prove that
H is a normal subgroup G if and only if Hg = gH for every g that exist G:

(Note: Hg = {hg : h exist in Hg} and gH = {gh : h exist in Hg} are sets. So, in the proof of -->, you suppose that H is a normal subgroup of G and then prove two inclusions: Hg is a subset of gH and gH is a subset of Hg.)

2. Originally Posted by mathgirl1188
Let H be a subgroup of a group G. Prove that
H is a normal subgroup G if and only if Hg = gH for every g that exist G:

(Note: Hg = {hg : h exist in Hg} and gH = {gh : h exist in Hg} are sets. So, in the proof of -->, you suppose that H is a normal subgroup of G and then prove two inclusions: Hg is a subset of gH and gH is a subset of Hg.)

$H\triangleleft G\Longleftrightarrow \forall g\in G\,,\,g^{-1}Hg=H\Longleftrightarrow gH=Hg$

Tonio