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MTH633 : Group Theory

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Dr. Imran Ahmed
Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan

Course Contents

 Properties of real and complex numbers, Binary Operation, Bijective maps, Inversion Theorem, Isomorphic binary structures, Groups, Examples of Groups, Uniqueness of identity and inverse, Elementary properties of groups, Abelian Groups, Order of a Group, Finite Groups, Subgroups, Examples of subgroups, Two Step subgroup test, One Step subgroup test, Cyclic Groups,

Permutation Groups, Examples of Permutation Groups, Theorem on permutation groups, Cayley''s Theorem, Orbits, Cycles, Disjoint Cycles, Cycle Decomposition, Parity of permutations, Alternating Group, Direct product, Finitely generated abelian groups, Cosets, Partition of group 

Lagrange''s Theorem, Applications of Lagrange''s theorem, Indices of subgroups, Converse of Lagrange''s Theorem, Homomorphism of Groups, Properties of homomorphism, Normal Subgroups, Morphism Theorem for groups, Application of Morphism theorem, properties of homomorphism, Normality of kernel of homomorphism, Normal group, Factor group, Cosets multiplcation & Normality

Examples on kernel of homomorphisms and group homomorphism, Factor group from homomorphism, Kernel of an injective homomorphism, Factor groups from Normal subgroups, Example of Morphism theorem of groups, Normal groups and Inner Automorphism, Factor group computations

Simple Group, Maximal Normal Subgroups, The Centre subgroup, Example of the Centre subgroup, Commutator subgroup, Generating set, Commutator subgroup, Automorphisms, Group Action on set, Stablizer, Orbits, Conjugacy and G-sets

Isomorphism Theorem, Second Isomorphism Theorem, Third Isomorphism Theorem, First Sylow Theorems, Second Sylow Theorems, Third Sylow Theorems, Application of Sylow Theorems