Normal subgroup
A normal subgroup of a group
is a subgroup of
for which the relation "
" of
and
is compatible with the law of composition on
, which in this article is written multiplicatively. The quotient group of
under this relation is often denoted
(said, "
mod
"). (Hence the notation
for the integers mod
.)
Description
Note that the relation is compatible with right multiplication for any subgroup
: for any
,
On the other hand, if
is normal, then the relation must be compatible with left multiplication by any
. This is true if and only
implies
Since any element of
can be expressed as
, the statement "
is normal in
" is equivalent to the following statement:
- For all
and
,
,
which is equivalent to both of the following statements:
- For all
,
;
- For all
,
.
By symmetry, the last condition can be rewritten thus:
- For all
,
.
Examples
In an Abelian group, every subgroup is a normal subgroup.
Every group is a normal subgroup of itself. Similarly, the trivial group is a subgroup of every group.
Consider the smallest nonabelian group, (the symmetric group on three elements); call its generators
and
, with
, the identity. It has two nontrivial subgroups, the one generated by
(isomorphic to
and the one generated by
(isomorphic to
). Of these, the second is normal but the first is not.
If and
are groups, and
is a homomorphism of groups, then the inverse image of the identity of
under
, called the kernel of
and denoted
, is a normal subgroup of
(see the proof of theorem 1 below). In fact, this is a characterization of normal subgroups, for if
is a normal subgroup of
, the kernel of the canonical homomorphism
is
.
Note that if is a normal subgroup of
and
is a normal subgroup of
,
is not necessarily a normal subgroup of
.
Group homomorphism theorems
Theorem 1. An equivalence relation on elements of a group
is compatible with the group law on
if and only if it is equivalent to a relation of the form
, for some normal subgroup
of
.
Proof. One direction of the theorem follows from our definition, so we prove the other, namely, that any relation compatible with the group law on
is of the form
, for a normal subgroup
.
To this end, let be the set of elements equivalent to the identity,
, under
. Evidently, if
, then
, so
; the converse holds as well, so
is equivalent to the statement "
". Also, for any
,
so
. Thus
is closed under the group law on
, so
is a subgroup of
. Then by definition,
is a normal subgroup of
.