Solve Group Theory Problem: Proving HuX is G

In summary, the conversation is about a problem involving normal subgroups and the subgroup generated by a subset of a group. The goal is to prove that the subgroup generated by the union of the normal subgroup and the subset is equal to the entire group. The person is trying to solve the problem directly and also considering the pullback of the subset. They plan to spend more time thinking about it and will have a meeting in 2 hours.
  • #1
JasonRox
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I'm stuck on this one...

I'm studying for my midterm so I'm solving problems for practice. Here's one of them...

Let H be a normal subgroup in G, and let v be the natural map from G to G/H, and let X be a subset of G such that the subgroup generated by v(X) is G/H. Prove that the subgroup generated by H union X (HuX) is G.

I'm trying to do this directly with showing if x is in G, then x is in <HuX> (generated subgroup). I tried doing contradiction too, by assuming <HuX> is some proper subgroup A of G and not G itself.

I'm going to spend more time thinking about this. I'll be back in like 2 hours since I have a meeting, which I'll spend a minute here or there thinking about it.
 
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  • #2
What's the pullback of <v(X)>?
 
  • #3
Good thought!

Not sure where it will lead me, but I'll think about that too.
 

Related to Solve Group Theory Problem: Proving HuX is G

1. What is group theory?

Group theory is a branch of mathematics that deals with the study of groups, which are sets of elements that follow specific operations and axioms. It is used to analyze the symmetries and structures of mathematical objects and can be applied in various fields, including physics, chemistry, and computer science.

2. What is a HuX group?

A HuX group is a type of finite group that has a special property known as the HuX property. This property means that every element of the group can be expressed as a product of two specific elements, one of which is the identity element. In other words, every element in a HuX group can be written as a combination of the identity element and another element in the group.

3. How do you prove that HuX is G?

To prove that HuX is G, we must show that all elements in the group can be expressed as a product of the identity element and another element in the group. This can be done by using the definition of a HuX group and showing that the property holds for all elements in the group.

4. What are some applications of group theory?

Group theory has many practical applications, including cryptography, coding theory, and particle physics. It is also used in the study of crystal structures, molecular orbitals, and symmetry in art and architecture.

5. Is group theory difficult to understand?

Group theory can be challenging for beginners, but with practice and patience, it can be understood by anyone with a strong foundation in algebra and abstract mathematical concepts. It is a highly specialized field, so it may require some effort to grasp the more complex ideas and applications.

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