# Characteristic subalgebras are ideals in the variety of groups

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, any characteristic subalgebra is an ideal.

### Translation to the language of groups

Every characteristic subgroup is a normal subgroup.

## Importance

Any subgroup-defining function, i.e. a function that picks a unique subgroup, must yield a characteristic subgroup. This result tells us that that characteristic subgroup must also be an ideal. Further, since the variety of groups is ideal-determined, there is a natural quotient map associated to this characteristic subgroup, giving rise to a quotient-defining function.

This leads to a bijection between subgroup-defining functions and quotient-defining functions: for every way of uniquely pinning down a subgroup, we have a way of uniquely pinning down a quotient map. For instance, the subgroup-defining function center corresponds to the quotient-defining function inner automorphism group.

## Proof

### Direct proof

The direct proof is by Facts (1) and (2).

### Underlying idea

The underlying idea here is that the ideal terms in the variety of groups are generated by terms of the form $\varphi(u,t) = tut^{-1}$, which also happen to be inner automorphisms when we view $t$ as a parameter and $u$ as the input variable. Thus, if a subgroup is invariant under all automorphisms, it is also closed under the ideal terms, and is hence an ideal.

This fails miserably in a number of other varieties. For instance, in the variety of commutative unital rings, the ideal terms are generated by terms of the form $\varphi(u,t) = ut$, $\varphi(u,t) = tu$, which are very far from ring automorphisms.