Brandeis MATH 101A - MATH 101A: ALGEBRA I PART A: GROUP THEORY (5 pages)

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MATH 101A: ALGEBRA I PART A: GROUP THEORY



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MATH 101A: ALGEBRA I PART A: GROUP THEORY

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Pages:
5
School:
Brandeis University
Course:
Math 101a - Precalculus, Algebra, Functions and Graphs

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18 MATH 101A ALGEBRA I PART A GROUP THEORY 7 Category theory and products I want to prove the theorem that a finite group is nilpotent if and only if it is the product of its Sylow subgroups For this we first have to go over the product of groups And this looks like a good time to introduce category theory 7 1 categories I gave the definition of a category and two examples to illustrate the definition Gps is the category of groups and Ens is the category of sets Definition 7 1 A category C consists of four things C Ob C M or id where 1 Ob C is a collection of objects This collection is usually not a set For example Ob Gps is the collection of all groups and Ob Ens is the collection of all sets 2 For any two objects X Y Ob C there is a set of morphisms M orC X Y which are written f X Y For example in M orGps G H is the set of homomorphisms G H and M orEns S T is the set of all mappings f S T 3 For any three objects X Y Z we have a composition law M orC Y Z M orC X Y M orC X Z sending g f to g f Composition must be associative 4 Every object X Ob C has an identity idX M orC X X so that idY f f f idX for any f X Y Note that there are only two assumptions about the structure Namely associativity of composition and the existence of units The idea of category theory is to extract elementary concepts out of difficult mathematics We look only at composition of morphisms and forget the rest of the structure Then we can ask What are the properties that can be expressed only in terms of composition of morphisms One of these is the product 7 2 product of groups Definition 7 2 If G H are groups then the product G H is defined to be the cartesian product of sets G H g h g G h H MATH 101A ALGEBRA I PART A GROUP THEORY 19 with the group law given coordinate wise by g1 h1 g2 h2 g1 g2 h1 h2 These are several things to notice about this definition The first is that G H contains a copy of G H which commute By this I mean that there are monomorphisms 1 1 homomorphisms j1 G G H j2 H G H



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