Normal subgroup equals kernel of homomorphism

From Groupprops

This article gives the statement, and possibly proof, of a basic fact in group theory.
View a complete list of basic facts in group theory
VIEW FACTS USING THIS: directly | directly or indirectly, upto two steps | directly or indirectly, upto three steps|
VIEW: Survey articles about this

This article gives a proof/explanation of the equivalence of multiple definitions for the term normal subgroup
View a complete list of pages giving proofs of equivalence of definitions

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:

  1. The map is well-defined, because if for , then (basically, we're using that ).
  2. 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