Group theory: Free groups

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I talk about a certain group-theoretic manifestation of general abstract nonsense, namely the notion of a free group. This is very similar in spirit to the notion of a free vector space on a set, or the tensor algebra T(V) of a vector space V (the latter is actually the free associative unital algebra on V).

I find the construction of the free group on a set to be a pretty straightforward idea, but I think it is often obscured by the omnipresent machinery of equivalence relations... so I try to give a brief overview of it. The general idea is that we treat everything formally. You should be able to sit down and make everything rigorous without too much thought.

Additional information: In both cases, we are concerned with existence/uniqueness of free objects in a certain category. The rigorous abstract definition of "free object" has to do with a functor which is left adjoint to the so-called "forgetful functor" U from C to Set. If we take the category C to be the category of groups (denoted Grp) where the morphisms are group homomorphisms, then U takes any group G to its underlying set, hence the name "forgetful" -- it "forgets" structure. The theory of adjoint functors is ubiquitous throughout mathematics, and is one of the most fruitful pearls of category theory.