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# ABSTRACT ALGEBRA I MATH 356

Rice University

GPA 3.86

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This 4 page Class Notes was uploaded by Jayde Lang on Monday October 19, 2015. The Class Notes belongs to MATH 356 at Rice University taught by Danijela Damjanovic in Fall. Since its upload, it has received 8 views. For similar materials see /class/224929/math-356-rice-university in Mathematics (M) at Rice University.

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Date Created: 10/19/15

Short summary for the rst miderm exam in Math 356 Spring 0 G is a group if gtk is closed on G associative there is an identity element and every element has an inverse If gtk is in addition commutative on G then G is abelian o G is a semigroup if gtk is closed on G associative G is a semigroup with unity if gtk is closed on G associa tive there is an identity element 0 Cancellation law holds in any group G ac hc implies a h and similarily cu ch implies a 17 G denotes cardinality of G G can be nite countably in nite or uncountably in nite G is nite if there are n elements in the set G where n is some natural number G is countably in nite if there is a bijection between G and N G is uncountably in nite if there is a bijection between G and R Examples of countably infinite sets are N Z uZ for some a E Z Q Examples of uncountably infinite sets are R any non empty interval in R set of irrational numbers There exists no hijection between a set which is countuhly infinite and one which is uncountuhly infinite There exists no hijection be tween ufinite set and an infinite set 0 G group H C G is a subgroup of G if it is a group on its own The easiest way to check if H lt G is to check if for all u h E H one has 01174 E H All subgroups of Z are of the form uZ for some a E Z Let G and H be two sets A map 1 G a H is onto if for every h E H there exists g E G such that 11 g h Map 1 is onetoone if 111g1 111gz for some g1g2 E G implies g1 g2 1 Map 1 is a bijection between two sets G and H if it is one to one and onto Let G and Ho be two groups A map 4 G H H is a group homomorphism if x y x 0 y for any two xy E G Every homomorphism must take identity in G to the iden tity in H and inverses in G to inverses in H Kernel ofahomomorphism Ker g E Gl g idH lt G Image ofahomomorphism 11114 4g E ng E G lt H lf Ker 6 then 4 is injective or one to one in which case we call 4 and embedding A homomorphism 4 is one to one if and only if j g id H for some g E G implies g idG If a homomorphism 4 is onto and one to one then it is called an isomorphism If in addition to this G H then 4 is an automorphism of G The set of all automorphisms of G has a group structure under the operation of composition of maps this group is denoted by AutG For a fixed a E G define the map Kg g H aga l These maps are called inner automorphisms of G InnG is the group of inner automorphisms ofG It is a subgroup of AutG G group Left multiplication in G by an element a of G is the map Ag g H agfor allg E G Cayley embedding theorem Each M is a bijection and if lGl n is finite the map A a H M is an embedding of G into Symn If H lt G then the image of H under the map M is called the left coset of H in G aH ahlh E Two cosets HH and bH are either disjoint or equal All cosets of H partition G into non overlapping pieces 3 o The set of all cosets of H in G aHla E G is denoted by GH o The number of left cosets of H in G is called the index of H in G and is denoted by G 0 Counting formulaLagrange theorem G H If G is a finite group and H lt G then H divides G and G H divides G and the index can be computed by GzH 0 Corollary 1 For g E G G finite order of g divides order of G ie ordg divides 0 Corollary 2Classi cation of groups of prime order Every group whose order is a prime number is cyclic 0 Corollary 3 If 4 G a H is a homomorphism then both Kergb and Imt divide G and Imgb divides H o A subgroup N lt G is normal in G if for all h E N and all g E G we have ghg 1 E N Equivalently N is normalin G if gNG 1 N Equivalently N is normal in G if all the left and right cosets gN and N g of N in G coincide Equivalently N is normal in G if N is invariant under all inner automorphisms of G o If N is a normal subgroup in G then the coset space G N has a group structure under the operation on cosets defined by aNbN abN o If 4 G a H is a homomorphism then Ker is a normal subgroup in G 0 First Isomorphism Theorem If 4 G a H is a homomor phism then the group GKergb is isomorphic to I 1114 In particular if 4 is onto then I mt H so we have G Ker is isomorphic to H 0 Classi cation of groups of order 2 where p is a prime num ber Every group of order 2 with p prirne is either cyclic or dihedral o G group For a E G the centralizer of a in G is the subgroup ZGa b E Glba ab 0 G group The center of G is the intersection of centralizers of all elements in G ZG b E Gllm abforalla E G Center of G is a normal subgroup of G o GZG is isomorphic to InnG o ZG G if and only if G is abelian o If G ZG is cyclic then it must be trivial and G must be abelian

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