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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-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