AlgTopReview4: Free abelian groups and non-commutative groups

Free abelian groups play an important role in algebraic topology. These are groups modelled on the additive group of integers Z, and their theory is analogous to the theory of vector spaces. We state the Fundamental Theorem of Finitely Generated Commutative Groups, which says that any such group is a direct sum of a finite number of Z (integers) together with a finite commutative group (the torsions part).

Non-commutative groups are also briefly introduced, mostly through the simplest example of S_3, the symmetric group on three objects, or alternatively the group of 3x3 permutation matrices. We illustrate that cosets of a subgroup in the commutative case are of two different types: left and right, and this makes the situation more complicated. When the left and right cosets agree we are in the situation of a normal group, and then the cosets do form a quotient group.

Thanks to Nguyen Le for filming.

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