Groups

Groups

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Examples of groups

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• Axioms for groups. Examples from geometry: symmetry groups of regular polygons, cube, tetrahedron. Permutations on a set; the symmetric group. Subgroups and homomorphisms. Symmetry groups as subgroups of general permutation groups. The M¨obius group; cross-ratios, preservation of circles, the point at infinity. Conjugation. Fixed points of M¨obius maps and iteration. [4]

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Lagrange’s theorem

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• Cosets. Lagrange’s theorem. Groups of small order (up to order 8). Quaternions. Fermat-Euler theorem from the group-theoretic point of view. [5]

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Group actions

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• Group actions; orbits and stabilizers. Orbit-stabilizer theorem. Cayley’s theorem (every group is isomorphic to a subgroup of a permutation group). Conjugacy classes. Cauchy’s theorem. [4]

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Quotient groups

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• Normal subgroups, quotient groups and the isomorphism theorem. [4]

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Matrix groups

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• The general and special linear groups; relation with the M¨obius group. The orthogonal and special orthogonal groups. Proof (in R3) that every element of the orthogonal group is the product of reflections and every rotation in R3 has an axis. Basis change as an example of conjugation. [3]

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Permutations

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• Permutations, cycles and transpositions. The sign of a permutation. Conjugacy in Sn and in An. Simple groups; simplicity of A5. [4]

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Appropriate books

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• M.A. Armstrong Groups and Symmetry. Springer–Verlag 1988(djvu)(pdf)
• † Alan F Beardon Algebra and Geometry. CUP 2005 R.P. Burn Groups, a Path to Geometry. Cambridge University Press 1987(pdf)
• J.A. Green Sets and Groups: a first course in Algebra. Chapman and Hall/CRC 1988
• W. Lederman Introduction to Group Theory. Longman 1976
• Nathan Carter Visual Group Theory. Mathematical Association of America Textbook(pdf)

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Recommended Books

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• Introduction to Group Theory Oleg Bogopolski(pdf)

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Informal description of Groups

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In university mathematics, algebra is the study of abstract systems of objects whose behaviour is governed by fixed rules (axioms). An example is the set of real numbers, which is governed by the rules of multiplication and addition. One of the simplest forms of abstract algebra is a group, which is roughly a set of objects and a rule for multiplying them together. Groups arise all over mathematics, particularly where there is symmetry.

The course introduces groups and their properties. The emphasis is on both the general theory and the many examples, such as groups of symmetries and groups of linear transformations.

Learning outcomes

By the end of this course, you should:

• be familiar with elementary properties of abstract groups, including the theory of mappings between groups;
• understand the group-theoretic perspective on symmetries in geometry.

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