Given a
mathematical group G and a
normal subgroup N of
G, the
factor group of
G over
N is the set of all the cosets (see under
subgroup) of
N in
G; it is denoted by
G/
N. Being
N normal in
G allows one to give in a natural way a group structure on
G/
N: if
aN and
bN are elements of
G/
N, the product
aN *
bN is by definition equal to
(ab)N, and by the normality of
N this definition is well-posed.
When G/N is finite, its order is equal to [G:N], the index of N in G.
Trivially, G/G is isomorphic to the group of order 1, and G/{1} is isomorphic to G.
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