Can a Finitely Generated Group Contain an Infinitely Generated Subgroup?

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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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