Beginners To an Advanced Course on Group Theory
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Topics that you will learn in this course
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Groups: Definition, Dihedral Groups, Symmetric Groups, Matrix Groups, The Quaternion Group, Cyclic Groups
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Subgroups: Definition, Centralizers, Normalizers, Stabilizer, Kernels, cyclic subgroups, subgroups generated by subsets of a group, the lattice of subgroups of a group,
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quotient groups, homomorphism, and Isomorphisms: Definition, cosets, Lagrange’s Theorem, The Isomorphism theorems, Transpositions, the Alternating Group
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Group Action: Group Actions and Permutation Representations, Groups Acting on themselves by Left Multiplication- Caley’s Theorem, Groups Acting on themselves by conjugation- The class equation, Automorphisms, The Sylow theorems, The simplicity of A_n.
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Direct and Semidirect products and Abelian Groups, The fundamental theorem of finitely generated abelian groups.
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Further Topics: p-group, Nilpotent Groups, and Solvable Groups.
Reference Books:
- Abstract Algebra by David S. Dummit and Richard M. Foote
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TOPICS IN ALGEBRA by Israel Nathan Herstein
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Contemporary abstract algebra by Joseph Gallian
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