Can a Finitely Generated Group Contain an Infinitely Generated Subgroup?

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In summary, a finitely generated group is a mathematical object with a finite number of elements and operations, following certain rules and properties. They differ from other groups by having a finite number of elements and can be expressed as a combination of a finite set of generators. Common examples include cyclic groups, dihedral groups, and finite abelian groups, and they are used in various fields of mathematics, such as algebra and topology. However, not all groups are finitely generated, but many important and interesting ones are.
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Could a finitely generated group contain a subgroup which is infinitely generated? Why?
 
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What does it mean that a group G is finitely generated? Can you give a definition?
What does a subgroup of G look like?
 
  • #3
Consider the free group generated by two elements a and b. I think you should be able to show that the subgroup generated by terms of the form bnab-n is not finitely generated.
 

1. What is a finitely generated group?

A finitely generated group is a mathematical object that is made up of a finite number of elements and operations. These elements and operations follow certain rules and properties, which allow them to be combined and manipulated in various ways.

2. How are finitely generated groups different from other groups?

Finitely generated groups are different from other groups in that they have a finite number of elements, whereas other groups may have infinitely many elements. Additionally, the elements of a finitely generated group can be expressed as a combination of a finite set of generators, while other groups may not have such a simple form.

3. What are some common examples of finitely generated groups?

Some common examples of finitely generated groups include cyclic groups, which are generated by a single element, and dihedral groups, which are generated by two elements. Other examples include finite abelian groups and certain types of free groups.

4. How are finitely generated groups used in mathematics?

Finitely generated groups are used in mathematics to study and understand various structures and properties. They can be applied to fields such as algebra, geometry, and topology, and have real-world applications in areas such as cryptography and computer science.

5. Are all groups finitely generated?

No, not all groups are finitely generated. Some groups, such as the group of real numbers under addition, have infinitely many elements and cannot be expressed as a combination of a finite set of generators. However, many important and interesting groups are finitely generated and are studied extensively in mathematics.

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