Smallest normal subgroup

1. Jan 21, 2008

ehrenfest

[SOLVED] smallest normal subgroup

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. 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?

2. Jan 21, 2008

Hurkyl

Staff Emeritus
"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)

Last edited: Jan 21, 2008
3. Jan 21, 2008

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.

4. Jan 21, 2008

ehrenfest

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

5. Jan 21, 2008

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.

6. Jan 21, 2008

Hurkyl

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

7. Jan 21, 2008

ehrenfest

But what is ______ in this case?

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

8. Jan 21, 2008

Hurkyl

Staff Emeritus
"the smallest normal subgroup that contains S"

9. Jan 21, 2008

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)?

10. Jan 21, 2008

Mathdope

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

11. Jan 21, 2008

Hurkyl

Staff Emeritus
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

Last edited: Jan 21, 2008