Normal subgroup
normal divisor, invariant subgroup
A subgroup of a group
for which the left decomposition of
modulo
is the same as the right one; in other words, a subgroup such that for any element
the cosets
and
are the same (as sets). In this case one also says that
is normal in
and writes
; if also
, one writes
. A subgroup
is normal in
if and only if it contains all
-conjugates of any of its elements (see Conjugate elements), that is
. A normal subgroup can also be defined as one that coincides with all its conjugates, as a consequence of which it is also known as a self-conjugate subgroup.
For any homomorphism the set
of elements of
that are mapped to the unit element of
(the kernel of the homomorphism
) is a normal subgroup of
, and conversely, every normal subgroup of
is the kernel of some homomorphism; in particular,
is the kernel of the canonical homomorphism onto the quotient group
.
The intersection of any set of normal subgroups is normal, and the subgroup generated by any system of normal subgroups of is normal in
.
Comments
A subgroup of a group
is normal if
for all
, or, equivalently, if the normalizer
, cf. Normalizer of a subset. A normal subgroup is also called an invariant subgroup because it is invariant under the inner automorphisms
,
, of
. A subgroup that is invariant under all automorphisms is called a fully-invariant subgroup or characteristic subgroup. A subgroup that is invariant under all endomorphisms is a fully-characteristic subgroup.
References
[a1] | M. Hall jr., "The theory of groups" , Macmillan (1959) pp. 26 |
[a2] | C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) pp. 5 |
[a3] | A.G. Kurosh, "The theory of groups" , 1 , Chelsea (1955) pp. Chapt. III (Translated from Russian) |
Normal subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_subgroup&oldid=21397