Normal subgroup equals kernel of homomorphism
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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Contents
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