Complement to normal subgroup is isomorphic to quotient group
From Groupprops
Contents
Statement
Suppose is a group,
is a normal subgroup and
is a subgroup such that
and
are permutable complements:
and
is trivial. Then,
.
In particular, any two permutable complements to are isomorphic to each other.
Related facts
Caveats
Note the following:
- If
is a normal subgroup of
, it is not necessary that
has a permutable complement in
. For instance, if
is the cyclic group of order four and
is a subgroup of order two,
does not have a permutable complement. A normal subgroup that does have a permutable complement is termed a complemented normal subgroup. A subgroup that occurs as the permutable complement to a normal subgroup is termed a retract.
- Any lattice complement to
in
is also a permutable complement to
in
. This is because any normal subgroup is permutable: its product with any subgroup is a subgroup. Further information: Equivalence of definitions of complemented normal subgroup
- Complements to abelian normal subgroup are automorphic: If
is an abelian normal subgroup of
and
are two permutable complements to
in
, then there is an automorphism of
that is the identity map on
and sends
to
. In particular,
and
are automorphic subgroups.
- Complements to normal subgroup need not be automorphic: If
is a normal subgroup of and
are two permutable complements to
in
, there need not be an automorphism of
sending
to
.
- Retract not implies normal complements are isomorphic: This states that if
and
are two permutable complements to a subgroup
, and both
and
are normal, this does not imply that
is isomorphic to
. In other words, interchanging the role of which subgroup is the normal one renders the result false.
- Schur-Zassenhaus theorem (normal Hall implies permutably complemented and Hall retract implies order-conjugate): This states that every normal Hall subgroup, and hence every normal Sylow subgroup, has a complement, and that any two such complements are conjugate subgroups in the whole group.
- Every group of given order is a permutable complement for symmetric groups: Any group of order
occurs as a permutable complement to
in
, via the embedding obtained by Cayley's theorem. This is a far cry from the fact that any two permutable complements to a normal subgroup are isomorphic.
- Semidirect product is not left-cancellative for finite groups: We can have finite groups
such that
, but
is not isomorphic to
. Note that the isomorphism between
and
must not map the normal subgroup
to the normal subgroup
, otherwise we would contradict the result of this page.