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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.
A normal subgroup can also be defined by: A subgroup N of a group G is a normal subgroup if N is a
union of conjugacy classes of 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.
All subgroups N of an abelian group G are normal,
because g-1Ng = g-1gN = N.
See also: characteristic subgroup
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