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