Can a Finitely Generated Group Contain an Infinitely Generated Subgroup?

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SUMMARY

A finitely generated group can indeed contain an infinitely generated subgroup. A group G is defined as finitely generated if there exists a finite set of elements such that every element of G can be expressed as a combination of these generators. In the discussion, the free group generated by two elements, a and b, serves as an example, demonstrating that the subgroup generated by elements of the form bnab-n is not finitely generated, thus confirming the existence of infinitely generated subgroups within finitely generated groups.

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  • Understanding of group theory concepts, specifically finitely generated groups.
  • Familiarity with subgroup definitions and properties.
  • Knowledge of free groups and their generation.
  • Basic grasp of algebraic structures in mathematics.
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  • Research the properties of free groups and their applications in group theory.
  • Explore the concept of subgroup generation and its implications in algebra.
  • Study examples of finitely generated groups containing infinitely generated subgroups.
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Mathematicians, students of abstract algebra, and anyone interested in advanced group theory concepts, particularly those exploring the relationships between finitely and infinitely generated groups.

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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?
 
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.
 

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