Proving Commutativity in Normal Subgroups with Abelian Subgroup Problem

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In summary, the conversation discusses how to show that NM is abelian by using the fact that N and M are normal subgroups of G and share only the identity element. The key to solving this is to look at the commutator nmn^(-1)m^(-1) and use the properties of normal subgroups to show that it equals the identity. Overall, the conversation provides a helpful example of how to approach problems involving normal subgroups.
  • #1
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Homework Statement
Suppose that N and M are two normal subgroups of G and that N and M share only the identity element. Show that for any n in N and m in M, nm = mn.

The attempt at a solution
I basically have to show that NM is abelian. Since N and M are normal, it follows that

[tex]nm = mn_1 = n_1m_1 = m_1n_2 = \cdots,[/tex]

where [itex]n_1,n_2,\ldots \in N[/itex] and [itex]m_1,m_2,\ldots \in M[/itex]. This is all I know. I don't know how to use the fact that [itex]N \cap M = \{e\}[/itex] or how it plays any role. Any tips?
 
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  • #2
Sure. Stare at n*m*n^(-1)*m^(-1) for a little bit. If you still don't see it, I'll give you another hint.
 
  • #3
Since M is normal, nmn-1 equals some m' in M and since N is normal, mn-1m equals some n' in N. This yield that m'm-1 = nn'. Aha! Then m'm and nn' must be the identity so m' is the inverse of m and n' is the inverse of n and after some rearrangement, we get that nm = mn.

Nice. What made you think of looking at nmn1m-1?
 
  • #4
I thought you'd get it written in that form. nm=mn is the same thing as nmn^(-1)m^(-1)=e. It even has a name, that expression is called the commutator. I think of commuting, I think of commutator. Just experience, I guess.
 

1. What is an Abelian Subgroup Problem?

An Abelian Subgroup Problem is a mathematical problem that involves finding a subgroup of a given group that is abelian, meaning that the elements of the subgroup commute with each other. This problem is important in group theory and has applications in various areas of mathematics, including algebra, geometry, and number theory.

2. What is the significance of solving an Abelian Subgroup Problem?

Solving an Abelian Subgroup Problem can help to deepen our understanding of group theory and its applications. It can also provide insights into the structure and properties of groups, and can be used to solve other mathematical problems.

3. How is an Abelian Subgroup Problem typically approached?

There are various approaches to solving an Abelian Subgroup Problem, depending on the specific group and its properties. One common method is to use the properties of abelian groups to identify potential subgroups, and then use other techniques, such as coset enumeration, to verify if these subgroups are indeed abelian.

4. Can an Abelian Subgroup Problem have multiple solutions?

Yes, an Abelian Subgroup Problem can have multiple solutions. This is because there can be more than one subgroup of a given group that is abelian. In fact, there are often infinitely many possible abelian subgroups, making the problem more challenging to solve.

5. Are there any real-world applications of the Abelian Subgroup Problem?

Yes, the Abelian Subgroup Problem has applications in various areas of mathematics, but it also has practical applications in other fields. For example, it can be used to study the symmetries of molecules in chemistry and the symmetries of crystals in materials science. It also has applications in cryptography, where abelian groups are used to create secure encryption algorithms.

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