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Example text

How many different bijections are there between a set S with n elements and itself? 2. 1. You may assume that given a family of disjoint subsets of a set, there is a way to choose one element in each member of the family13. 3. Prove that the inverse of a bijection is a bijection, and that the composition of two bijections is a bijection. 4. 1] Prove that ‘isomorphism’ is an equivalence relation (on any set of sets). 5. 3, for epimorphisms and surjections. 6. 4, explain how any function f : A → B determines a section of πA .

With notation as above: • Each identity 1A is an isomorphism, and is its own inverse. • If f is an isomorphism, then f −1 is an isomorphism, and further (f −1 )−1 = f . • If f ∈ HomC (A, B), g ∈ HomC (B, C) are isomorphisms, then the composition gf is an isomorphism, and (gf )−1 = f −1 g −1 . Proof. These all ‘prove themselves’. For example, it is immediate to verify that f −1 g −1 is a left-inverse of gf : indeed20, (f −1 g −1 )(gf ) = f −1 ((g −1 g)f ) = f −1 (1B f ) = f −1 f = 1A The verification that f f −1 −1 g .

11] Chapter II Groups, first encounter In this chapter we introduce groups, we observe they form a category (called Grp), and we study ‘general’ features of this category: what are the monomorphisms, epimorphisms in this category? what is the appropriate notion of ‘equivalence relation’ and ‘quotients’ for a group? does a ‘decomposition theorem’ hold in Grp? and other analogous questions. In Chapter III we will acquire a similar degree of familiarity with rings and modules. A more object-oriented analysis of Grp (for example, a treatment of the famous Sylow theorems, ‘composition series’, or the classification of finite abelian groups) is deferred to Chapter IV.

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Algebra, Chapter 0, Version 2009.01.18 by Aluffi P.

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