What is the Normalizer of a Subgroup?

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In summary, the number of distinct conjugate subgroups of a subgroup H in a group G is [G:N(H)] where N(H) is the centralizer of H. This can be proved using the counting formula |G|/|N(H)|=|C(H)|, where |C(H)| is the number of distinct conjugate subgroups H in G. It should be noted that this only applies to finite groups, and further exploration is needed for infinite groups. It may be helpful to seek assistance in the Linear Algebra thread, as this question has been answered there before.
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Homework Statement



Prove the number of distinct conjugate subgroups of a subgroup H in a group G is [G:N(H)] where N(H)={g [itex]\in[/itex] G | gHg[itex]^{-1}[/itex]=H}.

Homework Equations



I'm thinking the counting formula; G=|C(x)||Z(x)| with C(x) being the conjugacy class of x and Z(x) being the centralizer of x.

The Attempt at a Solution



I thought that I could say N(H) is the centralizer of H and [G:N(H)]=|G|/|N(H)| so by the counting formula |G|/|N(H)|=|C(H)| and |C(H)| is the number of distinct conjugate subgroups H. In truth I have only worked with the counting formula for elements, never with subgroups. Also if this does hold somewhat true, it seems to me to only apply to the case when G is finite. What about when G is infinite?
 
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Should I post this in the Linear Algebra thread? The rules say not to but it seems to me that questions such as this get answered there. I'm really just looking for if my solution is valid or not as well as a hint for the infinite case..
 

1. What is the definition of the normalizer of a subgroup?

The normalizer of a subgroup is the set of elements in a group that commute with all elements in the subgroup. In other words, it is the largest subgroup of the group that contains the given subgroup and also normalizes it.

2. How is the normalizer of a subgroup related to the centralizer of a subgroup?

The normalizer of a subgroup is a superset of the centralizer of the subgroup. While the normalizer contains all elements that commute with the subgroup, the centralizer only contains elements that commute with every element in the subgroup.

3. How can the normalizer of a subgroup be written in terms of the subgroup and the whole group?

The normalizer of a subgroup can be written as the intersection of the subgroup and its conjugates. In other words, it is the set of all elements in the group that can be written as gsg⁻¹, where g is an element of the group and s is an element of the subgroup.

4. What is the significance of the normalizer of a subgroup in group theory?

The normalizer of a subgroup is an important concept in group theory as it helps in understanding the structure of a group. It can be used to prove various theorems, such as the Sylow theorems, and has applications in many areas of mathematics, including algebra and number theory.

5. Can the normalizer of a subgroup be a subgroup itself?

Yes, the normalizer of a subgroup is always a subgroup of the whole group. This is because it contains the given subgroup and is closed under the group operation. Furthermore, the normalizer of a subgroup is the largest subgroup that contains the given subgroup and also normalizes it.

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