Ideals are subalgebras in the variety of groups

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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

In the variety of groups, viewed as a variety of algebras with zero, every ideal is a subalgebra.

Translation to the language of groups

In the language of groups, this is the statement that any normal subgroup is a subgroup.

Facts used

  • Variety of groups is ideal-determined: This proves, among other things, that normal subgroups are precisely the same thing as ideals in the variety of group with the identity element as the zero operation.

Proof

Underlying idea

The underlying idea is that the identity element of the group is a subgroup -- it is closed under all the group operations. Hence, any ideal must model the behavior of the identity element, and in particular, it must also be closed under all the group operations.

This is precisely what fails in other varieties of algebras with zero. For instance, in the variety of commutative unital rings, the zero element doesn't form a subalgebra, because it is not closed under the constant operation 1 (i.e., it does not contain the multiplicative identity element). That is why the ideals, which are modeled over the zero element, need not be unital subrings.