normal subgroup

#algebra

Definition

Suppose $H$ is a subgroup of group $G$ , then, $H$ is a normal subgroup if for all $h$ in $H$ , all $g$ in $G$ , $ghg^{-1}$ is in $H$ .

Proposition

above definition is equivalent to the following conditions:

for all $g$ in $G$ , $gHg^{-1} = H$
for all $g$ in $G$ , left coset $gH$ is equal to right coset $Hg$
every left coset of $H$ in $G$ is a right coset

References

M. Artin, Algebra, 2. ed. Boston, Mass. Munich: Pearson Education, Prentice Hall, 2011, p. 59.