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 Krishna Groups
 1 Algebraic Structure 2
 2 Group Definition 2
 3 Abelian Group
 4 Finite and infinite group 3
 5 General properties of a group
 6 Definitino of group based on left axioms 12
 7 Composition tables for finite sets.
 8 Addition modulo m 33
 9 Multiplication modulo p 34
 10 Residue classes of set of integers 41
 11 An alternative set of postulates for a group 47
 12 Permutations
 13 Groups of permutations 54
 14 Cyclic permutations 56
 15 Even and odd permutations 60
 16 Integral powers of an element of a group 74
 17 Order of an element of a group 75
 18 Homomorphism and Isomorphism of groups 92
 19 Properties of Homo and Isomorphis 100
 20 Relation of Isomorphism in set of all groups 108
 21 Transference of group structure 109
 22 Complexes and subgroups of a group 113
 23 Algebra of complexes
 24 Criteria for complexes to be subgroups 116
 THM1 2 step method; closed + inverse
 THM2 1 step method; H is subgroup if a,b \(\in\) H => ab \(\in\) H
 25 Criteria for product of subgroups to be subgroup 120
 26 Intersection of subgroups 120
 27 Cosets 130
 Properties 2 and 6
 28 Relation of congruence modulo 136
 29 Lagrange's Theorem 138
 30 Order of the product of two subgroups of finite order 141
 31 Cayleys Theorem 151
 32 Cyclic Groups 153
 Krishna Rings

 Rings 174

 Elementary properties of rings 175

 Integral multiples of elements of rings 176

 Examples of rings 179

 Some special types of rings 179

 Integral domain 181
 42 examples

 Isomorphism of rngs 216

 properties of isomorphism of rings 218

 Subrings 219
 If subset itelf is a ring then it is subring
 THM The necessary and sufficient conditno for non empty subset S of R to be subring for a,b \(\in\) S we have (a) ab \(\in\) S and (b) ab \(\in\) S
 ie. it is closed under subtration and multiplication

 Subfields 224

 Characteristics of ring 228

 Ordered Integral Domains 231

 Ideals 236
 21 egs

 More about ideals 251

 Ideals generated by a given subset of a ring 252

 Principal Ideal ring 254
 a comm ring without zero divisors and with unity is prinicpal ideal ring if every ideal S in R is principal ideal ie all S = (a)

 Divisibility in an integral domain 255
 Units = are those elements of ring which possess multiplicative inverse

 Polynomial rings 259
 3 egs

 Degree of the sum and product of two polynomials 263

 Rings of polynomials 264

 Polynomials over integral domain 265

 Polynomials over a field 266
 3 egs

 Krishna Rings Continued

 Divisibility of polynomials over a field

 Division algo for polynomials over a field
 1 eg

 Euclidean algo for polynomials over a field

 Unique factorization domain 6

 Unique factorization thm for polynomials over a field 7

 Value of polynomial at x = c

 Quotient Rings or Rings of Residue Classes 17

 Homo of rings 19

 Kernel of Ring homo 20
 4 egs

 Maximal Ideal 24

 Some more results on ideals 25

 Prime Ideals 28
 Let R be a ring and S an ideal in R ; then S is said to be a prime ideal if ab \(\in\) S , where a,b \(\in\) R => a \(\in\)S or b \(\in\) S
 R/S is ID iff S is prime ideal
 R is comm ring with unity => maximal ideals are prime ideals

 Euclidean Rings and Euclidean Domain 33

 Properties of Euclidean Rings 36

 Krishna Normal Subgroups

 Normal Subgroups

 Conjugate elments 63

 Conjugate subgroups 70

 Quotient groups 71

 Homo of groups 77

 Kernel of homo 80

 More reuls on group homo 87
