Correspondence Theorem for Groups .... Another Question ....

In summary: Joseph has a quick answer for him. Steenis offers some help. Joseph reflects on what was written, and ends the summary.
  • #1
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I am reading the book: "Advanced Modern Algebra" (Second Edition) by Joseph J. Rotman ...

I am currently focused on Chapter 1: Groups I ...

I need help with an aspect of the proof of Proposition 1.82 (Correspondence Theorem) ...

Proposition 1.82 reads as follows:
https://www.physicsforums.com/attachments/7995
View attachment 7996
In the above proof by Rotman we read the following:

" ... ... To see that \(\displaystyle \Phi\) is surjective, let \(\displaystyle U\) be a subgroup of \(\displaystyle G/K\). Now \(\displaystyle \pi^{-1} (U)\) is a subgroup of \(\displaystyle G\) containing \(\displaystyle K = \pi^{-1} ( \{ 1 \} )\), and \(\displaystyle \pi ( \pi^{-1} (U) ) = U\) ... ... "My questions on the above are as follows:
Question 1

How/why is \(\displaystyle \pi^{-1} (U)\) is a subgroup of \(\displaystyle G\) containing \(\displaystyle K\)? And further, how does \(\displaystyle \pi^{-1} (U) = \pi^{-1} ( \{ 1 \} )\) ... ... ?
Question 2

How/why exactly do we get \(\displaystyle \pi ( \pi^{-1} (U) ) = U\)? Further, how does this demonstrate that \(\displaystyle \Phi\) is surjective?
Help will be much appreciated ...

Peter
 
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  • #2
Q1: if $f:G \longrightarrow H$ is a group-homomorphism. If $B$ is a subgroup of $H$ then $\pi^{-1}B$ is a subgroup of $G$, if $A$ is a subgroup of $G$, then $\pi A$ is a subgroup of $H$. You can prove that easily.
Now if $k \in K=\pi^{-1}1$, then $\pi k =1 \in U$, and so on ...

Q2: In set-theory we have this property: Let $X$ en $Y$ be sets, and $f:X \longrightarrow Y$ is a function, if $B \subset Y$, then $ff^{-1}B \subset B$. If, furhermore, $f$ is surjective, then $ff^{-1}B = B$. Conversely, you can easily prove that if $ff^{-1}B = B$ then $f$ is surjective.
You know that $\pi$ is surjective. You can apply this now to your problem.
 
Last edited:
  • #3
steenis said:
Q1: if $f:G \longrightarrow H$ is a group-homomorphism. If $B$ is a subgroup of $H$ then $\pi^{-1}B$ is a subgroup of $G$, if $A$ is a subgroup of $G$, then $\pi A$ is a subgroup of $H$. You can prove that easily.
Now if $k \in K=\pi^{-1}1$, then $\pi k =1 \in U$, and so on ...

Q2: In set-theory we have this property: Let $X$ en $Y$ be sets, and $f:X \longrightarrow Y$ is a function, if $B \subset Y$, then $ff^{-1}B \subset B$. If, furhermore, $f$ is surjective, then $ff^{-1}B = B$. Conversely, you can easily prove that if $ff^{-1}B = B$ then $f$ is surjective.
You know that $\pi$ is surjective. You can apply this now to your problem.
Hi Steenis ... great to hear from you again ...

Thanks for your help ...

... just now reflecting on what you have written ...

Peter
 

1. What is the Correspondence Theorem for Groups?

The Correspondence Theorem for Groups, also known as the Fourth Isomorphism Theorem, is a fundamental theorem in group theory that states the relationship between the subgroups and quotient groups of a given group. It provides a way to understand the structure of a group by examining its subgroups and quotient groups.

2. How does the Correspondence Theorem for Groups work?

The Correspondence Theorem for Groups states that for any normal subgroup N of a group G, there exists a one-to-one correspondence between the subgroups of G containing N and the subgroups of G/N. This means that every subgroup of G/N corresponds to a unique subgroup of G containing N, and vice versa. This correspondence preserves the group structure, making it a useful tool in understanding the relationship between subgroups and quotient groups.

3. What is the significance of the Correspondence Theorem for Groups in group theory?

The Correspondence Theorem for Groups is an important tool in understanding the structure and properties of groups. It allows us to study the relationship between subgroups and quotient groups, and to classify and identify different types of groups. It also provides a way to simplify complex group structures by breaking them down into smaller, more manageable subgroups.

4. Can the Correspondence Theorem for Groups be applied to non-normal subgroups?

No, the Correspondence Theorem for Groups only applies to normal subgroups. A normal subgroup is a subgroup that is invariant under conjugation by elements of the group. This means that for any normal subgroup N of a group G, gNg⁻¹ = N for all g in G. If a subgroup is not normal, then the correspondence between subgroups and quotient groups does not hold.

5. Are there any real-world applications of the Correspondence Theorem for Groups?

While the Correspondence Theorem for Groups is primarily used in abstract mathematics, it has applications in other fields as well. For example, it can be used in cryptography to analyze the structure of certain codes and ciphers. It also has applications in physics, particularly in the study of symmetry and group theory in quantum mechanics.

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