Group Theory

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• Derive the existence of groups of a specified small order
• Understand and use the concept of conjugacy
• Recall and apply Burnside's formula
• Understand and use the terms homomorphism and isomorphism
• Understand and use the properties of group actions
• Recall and use the definitions and properties of dihedral, symmetric and alternating groups
• Recall and apply Sylow’s Theorems to determine the structure of certain groups of small order
• Recall and use the definitions and properties of cossets and subgroups
• Recall and apply Lagrange's theorem
• Understand, use the properties of and manipulate permutations
• Verify group properties in particular examples

• Revision form Linear Algebra II covering topics such as: • Matrix and disjoint cycle notation for permutations, expressing elements as products of transpositions, definitions and basic lemmas of abstract group theory: the uniqueness of the identity element and inverses • Permutation groups ; Matrix groups G1(n), S1(n), O(n), SO(n) ; The group of intergers mod n. • Cayley tables of small Symmetric Groups, Groups of Symmetries, dihedral groups (of small order) • Powers of elements • Order and generators for groups, orders of elements – examples Isomorphism and examples • Cyclic groups and their classification. • Any symmetric group is generated by transpositions • Odd and even permutations and the Alternating group • Conjugacy and conjugacy classes; example: cycle structures in symmetric groups and the relationship with conjugacy classes. • Definition and examples of subgroups and Lagrange's theorem; examples including cyclic groups and the Symmetric Groups • Testing for a subgroup. Intersections of subgroups are subgroups • Isomorphisms, their properties and the classification of small groups • Direct products and the internal direct product theorem • Normal subgroups, centre of a group, Cosets and quotient groups • Homomorphisms, kernel and image;the First Isomorphism Theorem • Group actions and Cayley’s theorem • Orbit stabiliser theorem and applications; the action of the dihedral groups on geometric features of regular n-gons, the rotation group of a cube is isomorphic to the symmetric group on 4 letters. • Burnside’s formula and its applications to enumeration of orbits e.g, counting seating plans, necklaces coloured dice • Application of group actions to finite group theory: Class equation, finite p-groups, Sylow’s theorems and their applications.

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Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat

An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.