Main Page | See live article | Alphabetical index 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-1xg is still in N. The statement N is a normal subgroup of G is written: . 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 kernelss 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. See also: characteristic subgroup This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.