Proving Induced Homomorphism from Normal Subgroup

  • Context: Graduate 
  • Thread starter Thread starter JasonRox
  • Start date Start date
  • Tags Tags
    Algebra General
Click For Summary
SUMMARY

The discussion centers on proving the induced homomorphism from a normal subgroup N in group G to group H via a homomorphism f. The key task is to demonstrate that the induced homomorphism f*: G/N -> H is well-defined and that it satisfies the properties of a homomorphism. The specific formulation is f*(Na) = f(a), where Na represents the coset of a in G/N. This establishes the necessary conditions for the induced homomorphism to exist.

PREREQUISITES
  • Understanding of group theory, specifically normal subgroups.
  • Familiarity with homomorphisms and their properties.
  • Knowledge of cosets and quotient groups.
  • Basic proficiency in mathematical proofs and logic.
NEXT STEPS
  • Study the properties of normal subgroups in group theory.
  • Learn about the construction of quotient groups and their significance.
  • Explore the concept of induced homomorphisms in abstract algebra.
  • Review examples of homomorphisms and their kernels in various groups.
USEFUL FOR

This discussion is beneficial for students of abstract algebra, particularly those studying group theory, as well as educators and mathematicians interested in the properties of homomorphisms and normal subgroups.

JasonRox
Homework Helper
Gold Member
Messages
2,394
Reaction score
4
The actual question is irrelevant. What I need to know is what are they actually asking me to do!

I'll write the question. I can probably solve it myself. Maybe I'm just too tired to think. I will ask my prof. during seminar on Monday anyways, but I'd rather not because he would problem just solve it and I don't want that.

The question is...

Let N be normal in G and let f:G->H be a homomorphism whose kernel contains N. Show that f induces a homomorphism f*:G/N->H by f*(Na) = f(a).

Do I only need to show that f* is well-defined and that f* is in fact a homomorphism?
 
Physics news on Phys.org
Yup just show it's well defined and that it's a homomorphism.
 
Perfect. That's all I needed to know.
 

Similar threads

Undergrad Question about "A Group Epimorphism is Surjective"
  • · Replies 13 ·
Replies
13
Views
2K
Undergrad Meaning of "passage to the quotient"?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Differences between left vs right actions in some group theory questions
  • · Replies 1 ·
Replies
1
Views
1K
MHB 4 problems regarding automorphisms/homomorphisms
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Understanding Induced Maps: Types and Applications
  • · Replies 5 ·
Replies
5
Views
3K
Graduate Showing that a certain subgroup is the kernal of a homomorphism
  • · Replies 9 ·
Replies
9
Views
4K
Undergrad Trouble understanding an online solution to an exercise in Dummit & Foote
  • · Replies 21 ·
Replies
21
Views
2K
MHB Prove function is a homomorphism
  • · Replies 2 ·
Replies
2
Views
4K
MHB Is the Function f a Homomorphism for Symmetry Group of a Circle?
  • · Replies 6 ·
Replies
6
Views
3K
MHB Inequality related to number of p-Sylow subgroups
  • · Replies 2 ·
Replies
2
Views
2K