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# Group Theory I & II: Course Outline (MAT 465) For Group Theory II

In mathematics, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups.

## Course Objective

At the end of this course the students will be able to write mathematical proofs and reason of groups and introduction to representation theory. Tools which will be studied here would enable student to study complicated cases of Syllow’s group.

## Course Contents

Actions of Groups, Permutation representation, Equivalence of actions, Regular representation, Cosets spaces, Linear groups and vector spaces, Affine groupa and affine spaces, Transitivity and orbits, Partition of G-spaces into orbits, Orbits as conjugacy class Computation of orbits, The classification of transitive G-spaces Catalogue of all transitive G-spaces up to G-isomorphism, One-one correspondence between the right coset of Ga and the G-orbit, G-isomorphism between coset spaces and conjugation in G, Simplicity of A_5, Frobenius-Burnside lemma, Examples of morphisms, G-invariance, Relationship between morphisms and congruences, Order preserving one-one correspondences between congruences on Ω and subrroups H of G that contain the stabilizer Gα, The alternating groups, Linear groups, Projective groups, Mobius groups, Orthogonal groups, unitary groups, Cauchy’s theorem, P-groups, Sylow P-subgroups, Sylow theorems, Simplicity of An when n > 5. 