Variety of groups is ideal-determined: Difference between revisions
(New page: {{variety of groups property satisfaction}} ==Statement== ===Statement in universal algebraic language=== Consider the variety of groups as a variety of algebras with zero, where the ze...) |
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===Translation to the language of groups=== | ===Translation to the language of groups=== | ||
The ideals in the variety of groups with zero are precisely the same as the normal subgroups. Thus, the result states that: | The ideals in the variety of groups with zero are precisely the same as the normal subgroups (this requires some justification). Thus, the result states that: | ||
* Every normal subgroup occurs as the kernel of a homomorphism. {{further|[[Normal subgroup equals kernel of homomorphism]]}} | * Every normal subgroup occurs as the kernel of a homomorphism. {{further|[[Normal subgroup equals kernel of homomorphism]]}} | ||
* A surjective homomorphism is completely determined by its kernel. In fact, it can be identified with the usual quotient map for a normal subgroup, to the coset space. {{further|[[First isomorphism theorem]]}} | * A surjective homomorphism is completely determined by its kernel. In fact, it can be identified with the usual quotient map for a normal subgroup, to the coset space. {{further|[[First isomorphism theorem]]}} | ||
==Facts used== | |||
# [[Normal subgroup equals kernel of homomorphism]] | |||
# [[First isomorphism theorem]] | |||
==Proof== | |||
===Proof outline=== | |||
The proof has three steps: | |||
# Every ideal is a normal subgroup | |||
# Every normal subgroup is a kernel of a homomorphism (by fact 1) | |||
# A surjective homomorphism is completely determined by its kernel (by fact 2) | |||
To put these steps together, consider the system of implications: | |||
Kernel of a congruence <math>\implies</math> Ideal <math>\implies^{(1)}</math>Normal subgroup <math>\implies^{(2)}</math> Kernel of a congruence | |||
Thus, all the notions ar eequivalent. (3) then completes the proof. | |||
Revision as of 15:56, 26 June 2008
This article gives the statement, and possibly proof, of a property satisfied by the variety of groups
View a complete list of such property satisfactions
Statement
Statement in universal algebraic language
Consider the variety of groups as a variety of algebras with zero, where the zero in a group is its identity element. Then, this variety is ideal-determined: the map that sends a congruence on a group to its kernel is a bijection from the set of congruences on the group to the set of ideals in the group. There are two statements being made:
- Every ideal occurs as the kernel of a congruence (or, as the kernel of a homomorphism)
- A congruence (or equivalently, a surjective homomorphism, or a quotient map) is completely determined by its kernel
Translation to the language of groups
The ideals in the variety of groups with zero are precisely the same as the normal subgroups (this requires some justification). Thus, the result states that:
- Every normal subgroup occurs as the kernel of a homomorphism. Further information: Normal subgroup equals kernel of homomorphism
- A surjective homomorphism is completely determined by its kernel. In fact, it can be identified with the usual quotient map for a normal subgroup, to the coset space. Further information: First isomorphism theorem
Facts used
Proof
Proof outline
The proof has three steps:
- Every ideal is a normal subgroup
- Every normal subgroup is a kernel of a homomorphism (by fact 1)
- A surjective homomorphism is completely determined by its kernel (by fact 2)
To put these steps together, consider the system of implications:
Kernel of a congruence Ideal Normal subgroup Kernel of a congruence
Thus, all the notions ar eequivalent. (3) then completes the proof.