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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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 given by j1 g g e j2 h e h These inclusion maps have commuting images since j1 g j2 h g e e h g h e h g e j2 h j1 g 7 2 1 internal direct product Lemma 7 3 If G K H K are homomorphisms with commuting images then there is a unique homomorphism f G H K so that f j1 and f j2 Proof This is obvious f must be given by f g h j1 g j2 h This is a homomorphism since j1 G j2 H e Theorem 7 4 Suppose that G contains normal subgroups H K so that H K e H K e and HK G Then the homomorphism f H K G given by the inclusion maps H G K G is an isomorphism We say that G H K is the internal direct product in this case Proof The map is given by f h k hk This is surjective since HK G It is 1 1 since H K e It is a homomorphism since H K e 7 2 2 universal property The product G H has two other projection homomorphisms p1 G H G p2 G H H given by p1 g h g p2 g h h These satisfy the following universal property which is obvious obviously true and which I also explained in categorical terms Theorem 7 5 Suppose that G H K are groups and K G K H are homomorphisms Then there exists a unique homomorphism f K G H so that p1 f and p2 f The unique homomorphism is f x x x and it is written f 20 MATH 101A ALGEBRA I PART A GROUP THEORY 7 3 categorical product The last theorem is categorical since it involves only composition of homomorphism It says that G H is a categorical product Definition 7 6 Suppose that X Y are objects of a category C Then Z C is the product of X and Y if there are morphisms p1 Z X p2 Z Y so that for any other object W and any morphisms W X W Y there is a unique morphism f W Z so that p1 f and p2 f The condition can be written as a commuting diagram X p1 f Z W p2 Y We say that Z is the product of X Y in the category C and we write Z X Y We also call Z the categorical product of X and Y Theorem 7 5 was written in such a way that it is obvious that the product of groups is the categorical product The next point I made was that the definition of a product defines Z X Y uniquely up to isomorphism The concept of an isomorphism is categorical Definition 7 7 Two objects X Y in any category C are isomorphic and we write X Y if there are morphisms f X Y and g Y X so that f g idY and g f idX The definition of product is by a universal condition which forces the object Z to be unique up to isomorphism if it exists If the product does not exist it suggests that the category is not large enough and perhaps we should add more objects Theorem 7 8 The product Z X Y is unique up to isomorphism assuming it exists Proof Suppose that Z is another product This means what we have morphisms p 1 Z X p 2 Z Y so that for any W such as W Z any morphisms W X W Y such as p1 p2 there is a unique morphism g so that p i g pi for i 1 2 In other words the MATH 101A ALGEBRA I PART A GROUP THEORY 21 following diagram commutes X g Z p 2 p 1 Y p1 p 2 Z Similarly since Z is the product there is a unique morphism f Z Z so that pi f p i for i 1 2 Now take Z and Z We have two morphisms Z Z making the following diagram commute X p1 idZ Z Z f g p2 p2 p1 Y By the uniqueness clause in the definition of the product we must have f g idZ Similarly g f idZ So Z Z 7 4 products of nilpotent groups Finally I proved the following theorem which we need I used three lemmas without proof But I am giving the proofs here after the proof of the theorem and retroactively in Corollary 5 …


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