Normal subgroup

In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element x in N and each g in G, the element g^-1 x g is still in N.

Another way to put this is saying that right and left cosets of N in G coincide:

N g = g g^-1 N g = g N for all g in G.

Normal subgroups are of relevance because if N is normal, then the factor group G/N may be formed. Normal subgroups of G are precisely the kernels of group homomorphisms f : G -> H.

{e} and G are always normal subgroups of G. If these are the only ones, then G is said to be simple.

		Normal subgroup
		In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element x in N and each g in G, the element g^-1 x g is still in N. Another way to put this is saying that right and left cosets of N in G coincide: N g = g g^-1 N g = g N for all g in G. Normal subgroups are of relevance because if N is normal, then the factor group G/N may be formed. Normal subgroups of G are precisely the kernels of group homomorphisms f : G `->` H. {e} and G are always normal subgroups of G. If these are the only ones, then G is said to be simple.
Find your way back!