Proving Normality of Subgroups in Cyclic Groups

In summary, the conversation discussed a homework question where the task was to prove that all subgroups of a group G are normal if all cyclic subgroups of G are normal. The individual asking for help mentioned their understanding of cyclic groups being abelian and their confusion about the question. The advice given was to try a proof by contradiction, where H is a subgroup of G assumed to be not normal, leading to a contradiction.
  • #1
Kuzu
15
0
I'm taking this course "abstract algebra" at university and I've been given some homework questions. I was able to solve all of them but one. And it would be great if anyone could help me with this.

The question is like this:
"If all cyclic subgroups of G are normal, then show that all subgroups of G are normal"


as I know all cyclic groups are abelian, and G itself is a subgroup of G so it is cyclic and abelian. Also I know that every subgroup of an abelian group is normal. So I didn't even understand the question completely.

How can I solve this?
 
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  • #2
Try a proof by contradiction. Take H to be a subgroup of G and suppose H is not normal. What does that tell you?
 

FAQ: Proving Normality of Subgroups in Cyclic Groups

1. What is Abstract Algebra?

Abstract Algebra is a branch of mathematics that deals with the study of algebraic structures such as groups, rings, and fields. It focuses on the abstract properties of these structures and their operations.

2. What is the purpose of Abstract Algebra?

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Abstract Algebra has numerous applications in various fields such as cryptography, coding theory, computer science, physics, and chemistry. It is also used in data analysis, image processing, and machine learning algorithms.

4. What are some key topics in Abstract Algebra?

Some key topics in Abstract Algebra include group theory, ring theory, field theory, and linear algebra. Other important topics include group actions, homomorphisms, substructures, and isomorphisms.

5. How should I approach solving Abstract Algebra homework questions?

It is important to understand the definitions and theorems in Abstract Algebra before attempting to solve homework questions. Make sure to practice and review examples to gain a better understanding of the concepts. It is also helpful to break down the question into smaller parts and use logical reasoning to solve each part step by step.

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