What Defines the Smallest Normal Subgroup Containing a Subset?

[ehrenfest]

[SOLVED] smallest normal subgroup

Homework Statement

Given any subset S of a group G, show that it makes sense to speak of the smallest normal subgroup that contains S. Hint: Use the fact that an intersection of normal subgroups of a group G is again a normal subgroup of G.

Homework Equations

The Attempt at a Solution

The hint makes the proof easy when G is finite. When G is infinite, I do not think that the result holds since the intersection, for example of two alpha_0 sets, can be the same cardinality of the original sets. Can someone confirm?

 

[Hurkyl]

"Smallest" is meant in the sense of the partial order given by the subgroup relation. It is not meant in the sense of the total preorder given by comparing cardinalities.

That said, in a preorder, it is perfectly okay for there to be more than one "smallest" element.


(eep! I hope the point of the exercise wasn't for you to discover this yourself)

 

[Mystic998]

I always understood it as a definition that the smallest set (possibly with restrictions) A containing a set B as the intersection of all sets (with the same restrictions) containing B.

 

[ehrenfest]

I am confused. What exactly do they want me to prove??

 

[Mystic998]

Good question. I hate these loosely worded problems about things "making sense."

Honestly, I'd just show that an arbitrary intersection of normal subgroups containing a nonempty set is a normal subgroup, and move on.

 

[Hurkyl]

"Does ____ makes sense?" often (usually?) means "Is ____ well-defined?"

 

[ehrenfest]

But what is ______ in this case?

I think I'll just take Mystic998's suggestion.

 

[Hurkyl]

"the smallest normal subgroup that contains S"

 

[ehrenfest]

And what exactly is your definition of "smallest" (I read post #2, but I want a real precise definition if you don't mind)?

 

[Mathdope]

I think you can take it to mean it's the intersection of all normal subgroups that contain S.

 

[Hurkyl]

And what exactly is your definition of "smallest" (I read post #2, but I want a real precise definition if you don't mind)?

As in any preorder, "smallest" is defined as follows:

Suppose that $\leq$ is a reflexive, transitive relation on a set P, so that $(P, \leq)$ is a preorder¹. X is a smallest element of $(P, \leq)$ if and only if, for every $Y \in P$, we have $X \leq Y$.

In this case, P is the set of subgroups containing S, and $\leq = \subseteq$.


I.E. an element is the smallest if and only if it is less than or equal to every element of your preordering.



1: It's a partial order if $\leq$ is also antisymmetric