Abstract Algebra: Commutative Subgroup

In summary, the conversation discusses proving that the set H, consisting of elements in a group G that commute with fixed elements a and b, is a subgroup of G. The subgroup test is used, and the main challenge is proving that the inverse of an element in H is also in H. The conversation suggests using the fact that axb = bxa and proving that [b-1,a] = [b,a-1] = 0 to solve this problem.
  • #1
pat bismark
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0

Homework Statement



Let G be a group and let a, b be two fixed elements which commute with each other (ab = ba). Let H = {x in G | axb = bxa}. Prove that H is a subgroup of G.


Homework Equations



None

The Attempt at a Solution



I'm using the subgroup test. I know how to show that the identity of G exists in H and that if x1, x2 exist in H then x1 times x2 exists in H but need help proving that if x is in H then x^-1 is also in H. I've tried starting with axb = bxa and multiplying on the right and left with various combinations of a, x^-1, and b, but can't get it to the form ax^-1b = bx^-1a.

Please let me know if you have any ideas, thanks.
 
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  • #2
Prove that [b-1,a] = [b,a-1] = 0 and use that along with the fact that axb = bxa.
 

Related to Abstract Algebra: Commutative Subgroup

1. What is a commutative subgroup in abstract algebra?

A commutative subgroup is a subset of a group that satisfies the two properties of closure and commutativity. Closure means that if two elements of the subgroup are combined using the group's operation, the result is also an element of the subgroup. Commutativity means that the order in which the elements are combined does not affect the result.

2. How is a commutative subgroup different from a regular subgroup?

A regular subgroup does not necessarily have to satisfy the property of commutativity, while a commutative subgroup specifically has to satisfy both the properties of closure and commutativity. In other words, a commutative subgroup is a special type of subgroup.

3. What are some examples of commutative subgroups?

Examples of commutative subgroups include the set of even numbers under addition, the set of nonzero rational numbers under multiplication, and the set of all rotations of a square under composition.

4. Why is the property of commutativity important in abstract algebra?

The property of commutativity allows for simpler and more general mathematical structures to be studied. Many familiar mathematical operations, such as addition and multiplication, are commutative, so studying commutative subgroups can help us understand these operations better.

5. How is the concept of a commutative subgroup used in real-world applications?

The concept of a commutative subgroup is used in various fields such as cryptography, coding theory, and physics. For example, in coding theory, commutative subgroups are used to study error-correcting codes, while in physics, they are used to study the symmetries of physical systems.

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