M1: Groups and Group Actions (2022-23)
Section outline
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Lectures 1-2 will cover the group axioms, Cayley tables and some basic examples of groups, e.g. cyclic groups, dihedral groups and matrix groups.
Lectures 3-4 will cover the symmetric group: permutations, cycle decomposition, transpositions, even and odd permutations and conjugacy.
Lectures 5-6 will cover subgroups, cyclic groups and the Chinese remainder theorem for cyclic groups.
Lectures 7-8 will cover equivalence relations, modular arithmetic, cosets and Lagrange's theorem.
Note that only the main course questions in the sheets are meant for submitting work and discussion in tutorials. The S (starter) and P (pudding) questions are optional and solutions will be made available. (Of course you are free to discuss them with your tutors if you wish.)
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This sheet covers the first two lectures: binary operations, the group axioms and examples of groups.
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This problem sheet covers permutations: transpositions, parity and conjugacy.
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This problem sheet covers subgroups, cyclic subgroups and equivalence relations.
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This sheet covers material on modular arithmetic, cosets and Lagrange's theorem.
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This sheet covers homomorphisms, conjugacy and normal subgroups.
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This sheet covers quotient groups and basics of group actions
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This sheet covers the Orbit-Stabilizer Theorem.
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This sheet covers applications of the Orbit-Counting formula and symmetries of regular polyhedra.