Correspondence Theorem in Group Theory

In summary, the theorem states that every subgroup of the quotient group is of the form ##H/N## for some subgroup ##H## of ##G## with ##H\le G##.
  • #1
TheShrike
44
1
Hello,

I'm following the proof for this theorem in my textbook, and there is one part of it that I can't understand. Hopefully you can help me. Here is the part of the theorem and proof up to where I'm stuck:

Let ##N## be a normal subgroup of a group ##G##. Then every subgroup of the quotient group ##G/N## is of the form ##H/N## for some subgroup ##H## of ##G## with ##H\le G##.

...

Proof: Let ##H^*## be a subgroup of ##G/N## so that it consists of a certain set ##\{hN\}## of left cosets of ##N## in ##G##. We define the subset ##\beta(H^*)## of ##G## to be ##\{g\in G:gN\in H^*\}##. Then ##\beta(H^*)## clearly contains N and is a subgroup of G:

[here follows demonstration that ##\beta(H^*)## is a subgroup of G]

The portion in red is what I'm having trouble with. I don't see why it's immediately clear that ##N## is contained within ##\beta(H^*)##.

Any help is appreciated.

I first thought that perhaps ##H^*## is supposed to correspond to the ##H## of the hypothesis, but then I realized that cannot be true, since the elements of ##\beta(H^*)## would be precisely those of ##H## and the properties would be trivial. In particular, proving that ##\beta(H^*)## is a subgroup of ##G## would be pointless.
 
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  • #2
Isn't the identity of ##H^*## exactly the coset ##N##?
 
  • #3
That's correct. So we know the identity element ##1## belongs to ##\beta(H*)##. We also know that ##N## is a subgroup, so ##1\in N##. Therefore ##\beta(H*)## contains at least one element of ##N##.
 
  • #4
TheShrike said:
...

We define the subset ##\beta(H^*)## of ##G## to be ##\{g\in G:gN\in H^*\}##.

For each [itex] g \in N ,\ gN = N[/itex] and [itex] N \in H^* [/itex], so [itex] g\in \beta(H^*) [/itex]
 
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Likes 1 person
  • #5
Ah, well that clears that up. Thanks.
 

1. What is the Correspondence Theorem in Group Theory?

The Correspondence Theorem, also known as the Fourth Isomorphism Theorem, is a fundamental result in group theory that relates the subgroups and quotient groups of a given group to its normal subgroups and quotient groups. It states that there is a one-to-one correspondence between the subgroups of a given group and the normal subgroups containing the kernel of a homomorphism from that group. This theorem helps in understanding the structure of a group and its various subgroups.

2. How is the Correspondence Theorem used in group theory?

The Correspondence Theorem is used to establish a relationship between the subgroups and normal subgroups of a given group. It helps to determine the number of subgroups and quotient groups of a given group, and also provides a way to construct new groups from known ones. This theorem is also used to prove other important results in group theory, such as the Isomorphism Theorem and the Lattice Isomorphism Theorem.

3. Can the Correspondence Theorem be applied to any group?

Yes, the Correspondence Theorem is a general result that can be applied to any group. It holds true for both finite and infinite groups, as well as for abelian and non-abelian groups. However, the theorem is most commonly used for finite groups as it provides a useful tool for understanding their structure and properties.

4. What is the relation between the Correspondence Theorem and the Isomorphism Theorem?

The Correspondence Theorem and the Isomorphism Theorem are closely related and are often used together in group theory. The Correspondence Theorem provides a way to construct new groups from known ones, while the Isomorphism Theorem states that two groups are isomorphic if and only if their corresponding subgroups are isomorphic. In other words, the Correspondence Theorem helps to establish the isomorphism between subgroups, while the Isomorphism Theorem establishes the isomorphism between groups.

5. How does the Correspondence Theorem relate to normal subgroups?

The Correspondence Theorem plays a crucial role in understanding normal subgroups. It states that every normal subgroup of a group G is the kernel of a homomorphism from G to another group. This means that normal subgroups can be identified as the kernels of homomorphisms, and this helps in determining the properties and structure of normal subgroups. Additionally, the Correspondence Theorem also establishes a relationship between normal subgroups and quotient groups, as normal subgroups are precisely the ones that correspond to the trivial subgroup in the quotient group.

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