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This question is partly inspired by the recent question on measurability and the axiom of choicemeasurability and the axiom of choice.

Suppose I come up with a way to define a subgroup of a group, in a way that involves "no arbitrary choices" in some sense. Then chances are, the subgroup will be normal. For example, the commutator subgroup of a group is normal, as is the commutator subgroup of the commutator subgroup.

I seem to recall that there are general theorems that say something like, any subgroup that is first-order definable without parameters is normal. Is this true? If so, are there even stronger theorems of this type?

The motivation is that such a meta-theorem could allow me, when I'm introducing a new kind of subgroup, to sidestep an explicit verification that the subgroup is normal.

This question is partly inspired by the recent question on measurability and the axiom of choice.

Suppose I come up with a way to define a subgroup of a group, in a way that involves "no arbitrary choices" in some sense. Then chances are, the subgroup will be normal. For example, the commutator subgroup of a group is normal, as is the commutator subgroup of the commutator subgroup.

I seem to recall that there are general theorems that say something like, any subgroup that is first-order definable without parameters is normal. Is this true? If so, are there even stronger theorems of this type?

The motivation is that such a meta-theorem could allow me, when I'm introducing a new kind of subgroup, to sidestep an explicit verification that the subgroup is normal.

This question is partly inspired by the recent question on measurability and the axiom of choice.

Suppose I come up with a way to define a subgroup of a group, in a way that involves "no arbitrary choices" in some sense. Then chances are, the subgroup will be normal. For example, the commutator subgroup of a group is normal, as is the commutator subgroup of the commutator subgroup.

I seem to recall that there are general theorems that say something like, any subgroup that is first-order definable without parameters is normal. Is this true? If so, are there even stronger theorems of this type?

The motivation is that such a meta-theorem could allow me, when I'm introducing a new kind of subgroup, to sidestep an explicit verification that the subgroup is normal.

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Timothy Chow
Timothy Chow
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Proving that a subgroup is normal

This question is partly inspired by the recent question on measurability and the axiom of choice.

Suppose I come up with a way to define a subgroup of a group, in a way that involves "no arbitrary choices" in some sense. Then chances are, the subgroup will be normal. For example, the commutator subgroup of a group is normal, as is the commutator subgroup of the commutator subgroup.

I seem to recall that there are general theorems that say something like, any subgroup that is first-order definable without parameters is normal. Is this true? If so, are there even stronger theorems of this type?

The motivation is that such a meta-theorem could allow me, when I'm introducing a new kind of subgroup, to sidestep an explicit verification that the subgroup is normal.