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.

Facts used

1. Characteristic implies normal
2. Normal subgroup equals kernel of homomorphism.

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.

Also see: group acts as automorphisms by conjugation.

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.