Symmetry groups and Caley tables

In summary, the conversation discusses the requirements for a homework assignment involving a shape with rotational symmetry but not reflectional symmetry. The task includes writing down the elements of the symmetry group in standard notation and constructing a Cayley table under composition of symmetries. The speaker clarifies their confusion with the terminology and asks for guidance on the definitions of "standard notation" and "Cayley table". The responder provides a definition for the Cayley table and suggests using the notation set in the course or textbook. The speaker expresses frustration with vague math books.
  • #1
benedwards2020
41
0

Homework Statement



I have a shape about the origin. It has rotational symmetry but not reflectional symmetry (its an odd star shape!).

I have to write down in standard notation the elements of the symmetry group and I have to construct a caley table under composition of symmetries.

I think I'm getting mixed up with some of the terminology here. By standard notation does it mean

S(Q) = {e, Rpi/2, Rpi,...) etc?

And the caley table under composition symmetries, does it mean simply the table constructed with each element across the top and down the side and then calculated in the normal way?

Thanks
 
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  • #2
The definition of 'Standard notation' is almost certainly purely local, and definitely not standard. Sorry. You need to use whatever notation has been set in your book or lecture course.

The definition of Cayley table is precisely what you wrote.
 
  • #3
Many thanx... Sometimes these maths books can be a bit vague
 

1. What are symmetry groups?

Symmetry groups refer to a mathematical concept that represents the collection of symmetries of an object or system. These symmetries can include rotations, reflections, translations, and other transformations that preserve the overall shape and structure of the object.

2. How are symmetry groups classified?

Symmetry groups are classified based on the type of transformations that preserve the symmetry. This can include rotational symmetry groups, which consist of rotations around a fixed point, or reflectional symmetry groups, which consist of mirror reflections across a line or plane.

3. What is a Caley table?

A Caley table, also known as a multiplication table, is a mathematical tool used to organize and visualize the various combinations and results of a binary operation. In the context of symmetry groups, a Caley table can be used to represent the group's elements and how they combine to form the symmetry transformations.

4. How are Caley tables used in studying symmetry groups?

Caley tables are used to analyze the structure and properties of symmetry groups. By filling out the table with the group's elements and their combinations, patterns and relationships can be observed, which can provide insight into the group's properties and symmetries.

5. What is the significance of symmetry groups in science?

Symmetry groups have numerous applications in science, including in physics, chemistry, and materials science. They can help explain the symmetries and properties of physical objects and systems, and have important implications for understanding the underlying laws and principles of the natural world.

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