Variety of groups is ideal-determined

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

Related facts

Facts about other algebraic structures

Similar algebraic structures that are ideal-determined:

• Variety of algebra loops is ideal-determined: The ideals in this case are normal subloops.
• Variety of Lie rings is ideal-determined: The ideals in this case are ideals in the sense of Lie rings.
• Variety of power-associative loops is ideal-determined
• Variety of alternative loops is ideal-determined

Similar algebraic structures that are not ideal-determined:

• Variety of monoids is not ideal-determined

Facts used

1. Normal subgroup equals kernel of homomorphism
2. First isomorphism theorem

Proof

Proof outline

The proof has three steps:

1. Every ideal is a normal subgroup
2. Every normal subgroup is a kernel of a homomorphism (by fact 1)
3. 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 $\implies$ Ideal $\implies^{(1)}$Normal subgroup $\implies^{(2)}$ Kernel of a congruence

Thus, all the notions ar eequivalent. (3) then completes the proof.