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.
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.
A first course in abstract algebra by