# Problem Sheet E

This sheet is based on Lectures 9 and 10.

- Problem 1 asks to you show that the abelianisation of a group can be defined by a universal property, and then asks you to verify that it is indeed given by quotienting out the commutator subgroup.
- Problem 2 asks you to prove that the Hurewicz map is
*natural*. The precise meaning of the word "natural" will be explained later on in the course when we study*natural transformations* - Problems 3 and 4 are both examples of how to "cancel" things out in homology.
*Warning*:*not* - Problem 5 shows that the definition of a short exact sequence of chain complexes is not nonsensical.
- Problem 6 is a generalisation of Problem 4 on Problem Sheet D.

ðŸ¤“ Feel free to ask a question if you are stuck! ðŸ¤“

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