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this lecture is part of an online math course on group theory and will be about three groups so well start by quickly reviewing free abelian groups and then see how much of this theory extends to free possibly non abelian groups so a free abelian group on various elements a b c and so on is a sort of universal a billion group generated by these elements a b c and so on so universal means it should be as big as possible so we we dont put on any unnecessary relations like insisting a should of order three and so on and its pretty obvious what the free abelian group on this is if weve got elements a b c in an abelian group then then the group subgroup generated then consists of all elements of form n 1 a plus n to be and so on where the ni r and z and that the most universal way of doing this is just to take a sum of copies of the group z so this Z is going to be generated by a this one by B and so on so the free abelian group and a finite set of generators is just a sum of copies of
