Definition of an Abelian Group
A definition of an abelian group is provided along with examples using matrix groups. The general linear group and the special linear group are introduced.
Units Modulo n
This video introduces the units modulo n and gives a sketch of a proof showing that they form a group under multiplication modulo n.
Matrix Groups
Matrices are a great example of infinite, nonabelian groups. Here we introduce matrix groups with an emphasis on the general linear group and special linear group. The general linear group is written as GLn(F), where F is the field used for the matrix elements. The most common examples are GLn(R) and GLn(C). Similarly, the special linear group is written as SLn.
Symmetries of an Equilateral Triangle
An exploration of the smallest nonabelian group: the group consisting of symmetries of an equilateral triangle!
Symmetry Groups of Triangles
We introduce the connection between geometric figures and abstract algebra by showing how you can associate a group with triangles. The more symmetric the triangle, the larger the group. We illustrate this by finding the group of symmetries for equilateral, isosceles and scalene triangles.
Symmetries of a Square
A description of the dihedral group D4 (sometimes called D8) consisting of the symmetries of a square.
Dihedral Group
The Dihedral Group is a classic finite group from abstract algebra. It is a non abelian groups (non commutative), and it is the group of symmetries of a regular polygon. This group is easy to work with computationally, and provides a great example of one connection between groups and geometry.
Group or Not Group? (Integer edition)
Play the game show to test your understanding of the definition of a group!