# Thread: Normal and Quotient groups

1. ## Normal and Quotient groups

Let G be a group
H be a subgroup of G

G|H be set of all right cosets of H in G

Is it correct to say H is normal sub-group of G, IFF (IF AND ONLY IF) G|H is a group?

Thanks,
Aman

2. Originally Posted by aman_cc
Let G be a group
H be a subgroup of G

G|H be set of all right cosets of H in G

Is it correct to say H is normal sub-group of G, IFF (IF AND ONLY IF) G|H is a group?

Thanks,
Aman
one side is probably in your textbook. the other side: if $G/H$ is a group, then $f: G \longrightarrow G/H$ defined by $f(g)=Hg$ is a well-defined group homomorphism and $\ker f = H.$ thus $H \lhd G.$