finitely generated groups...

[tgt]

Could a finitely generated group contain a subgroup which is infinitely generated? Why?

[CompuChip]

What does it mean that a group G is finitely generated? Can you give a definition?
What does a subgroup of G look like?

[gel]

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.