Proof
1 implies 2
2 implies 1
Suppose gH = Hg.
Then for each h 
H
we have an element h' H
with
gh = h'g.
ghg-1 = h' H.
H
we have an element h' H
with
So,
H.
Suppose for all g G
and h H
we have
ghg-1 H.
G
and h H
we have
H.
Then gh can be written in the form h'g for some
h' H. This shows that
gH is contained in Hg.
Now apply this for g-1. Hence g-1H is contained in Hg-1. Multiplying by g from the left and by g-1 from the right yields that Hg is contained in gH.