What is "normal subgroup"

n. (context group theory English) A subgroup ''H'' of a group ''G'' that is invariant under conjugation; that is, for all elements ''h'' of ''H'' and for all elements ''g'' in ''G'', the element ''ghg''⁻¹ is in ''H''.

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup of a group is normal in if and only if for all in ; i.e., the sets of left and right cosets coincide. Normal subgroups (and only normal subgroups) can be used to construct quotient groups from a given group.

Évariste Galois was the first to realize the importance of the existence of normal subgroups.