Normal subgroup equals kernel of homomorphism
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This article gives the statement, and possibly proof, of a basic fact in group theory.
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This article gives a proof/explanation of the equivalence of multiple definitions for the term normal subgroup
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Statement
Verbal statement
A subgroup of a group occurs as the Kernel (?) of a group homomorphism if and only if it is normal.
Symbolic statement
A subgroup of a group
occurs as the kernel of a group homomorphism if and only if, for every
in
,
.
Definitions used
Kernel of a group homomorphism
A map is a homomorphism of groups if
-
for all
in
-
-
The kernel of is defined as the inverse image of the identity element under
.
Normal subgroup
For the purpose of this statement, we use the following definition of normality: a subgroup is normal in a group
if
contains each of its conjugate subgroups, that is,
for every
in
.
Related facts
Closely related to this are the isomorphism theorems.
Proof
Kernel of homomorphism implies normal subgroup
Let be a homomorphism of groups. We first prove that the kernel (which we call
) of
is a subgroup:
- Identity element: Since
,
is contained in
- Product: Suppose
are in
. Then
and
. Using the fact that
, we conclude that
. Hence
is also in
.
- Inverse: Suppose
is in
. Then
. Using the fact that
, we conclude that
. Hence,
is also in
.
Now we need to prove that is normal. In other words, we must show that if
is in
and
is in
, then
is in
.
Since is in
,
.
Consider . Hence,
must belong to
.
Normal subgroup implies kernel of homomorphism
Let be a normal subgroup of a group
. Then,
occurs as the kernel of a group homomorphism. This group homomorphism is the quotient map
, where
is the set of cosets of
in
.
The map is defined as follows:
Notice that the map is a group homomorphism if we equip the coset space with the following structure:
This gives a well-defined group structure because, on account of being normal, the equivalence relation of being in the same coset of
yields a congruence.
Explicitly:
- The map is well-defined, because if
for
, then
(basically, we're using that
).
- The image of the map can be thought of as a group because it satisfies associativity (
), has an identity element (
itself), has inverses (the inverse of
is
)
Further information: quotient map
References
Textbook references
- Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, More info, Page 82, Proposition 7