(ↄ)|Jai Ramanathan:Classes :Groups

Classes

Groups Part IA

Examples of groups

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]

Lagrange’s theorem

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

Group actions

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]

Quotient groups

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

Matrix groups

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]

Permutations

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

Appropriate books

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)

Introduction to Group Theory Oleg Bogopolski(pdf)

Associated Github Page https://jaircambridge.github.io/Groups/

Groups

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.

Past Example Sheets for this Course

Name	Latest commit message	Commit time
Group Theory Notes.pdf	Cambridge Part IA Groups	1 minute ago
Groups (2007-2008)(ex1).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2007-2008)(ex2).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2007-2008)(ex3).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2007-2008)(ex4).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2008-2009)(ex1).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2008-2009)(ex2).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2008-2009)(ex3).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2008-2009)(ex4).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2009-2010)(ex1).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2009-2010)(ex2).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2009-2010)(ex3).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2009-2010)(ex4).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2010-2011)(ex1).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2010-2011)(ex2).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2010-2011)(ex3).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2010-2011)(ex4).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2011-2012)(ex1).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2011-2012)(ex2).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2011-2012)(ex3).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2011-2012)(ex4).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2012-2013)(ex1).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2012-2013)(ex2).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2012-2013)(ex3).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2012-2013)(ex4).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2013-2014)(ex1).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2013-2014)(ex2).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2013-2014)(ex3).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2013-2014)(ex4).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2014-2015)(PracticeA).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2014-2015)(PracticeB).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2014-2015)(PracticeC).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2014-2015)(ex1).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2014-2015)(ex2).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2014-2015)(ex3).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2014-2015)(ex4).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2015-2016)(ex1).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2015-2016)(ex2).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2015-2016)(ex3).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2015-2016)(ex4).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2016-2017)(PracticeA).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2016-2017)(PracticeB).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2016-2017)(PracticeC).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2016-2017)(ex1).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2016-2017)(ex2).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2016-2017)(ex3).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2016-2017)(ex4).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2017-2018)(ex1).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2017-2018)(ex2).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2017-2018)(ex3).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2017-2018)(ex4).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2018-2019)(ex1).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2018-2019)(ex2).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2018-2019)(ex3).pdf	Cambridge Part IA Groups	1 minute ago
Groups (2018-2019)(ex4).pdf	Cambridge Part IA Groups	1 minute ago
Groups(McGill University).pdf	Cambridge Part IA Groups	1 minute ago