Are the Right and Left Cosets Equal in a Group's Cayley Table?

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In summary, there is a theorem that states that if a subgroup is normal, then the left and right cosets are equal, allowing for simplification of coset multiplication. However, if the subgroup is not normal, the result of multiplying cosets may not be a coset and cannot be simplified.
  • #1
mathsdespair
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Just by looking at the cayley table of a group and looking at its subgroups, is their a theorem or something which tells you if the right and left cosets are equal?

I have question to do and I would love to half the workload by not having to to work out the same thing twice.
Thanks
 
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  • #4
When multiplying cosets, can you just quite simply multiply them together?
Our teacher said something about their could be a potential problem?
What could the problem be?
 
  • #5
mathsdespair said:
When multiplying cosets, can you just quite simply multiply them together?
Our teacher said something about their could be a potential problem?
What could the problem be?
You can always multiply cosets, but the result is not necessarily a coset. In other words, the set of right or left cosets is not generally a group. Thus you can form the product ##aHbH##, which is the set of all elements of the form ##ah_1bh_2##, but this does not generally equal ##abH##, nor can it generally be written in the form ##gH## at all.

However, if ##H## is a normal subgroup, then the left and right cosets are the same (##aH = Ha##), so we just call them cosets, and the set of cosets does form a group. In this case, the product ##aHbH## can be simplified as follows:
$$aHbH = a(Hb)H = a(bH)H = abHH = abH$$
 
  • #6
jbunniii said:
You can always multiply cosets, but the result is not necessarily a coset. In other words, the set of right or left cosets is not generally a group. Thus you can form the product ##aHbH##, which is the set of all elements of the form ##ah_1bh_2##, but this does not generally equal ##abH##, nor can it generally be written in the form ##gH## at all.

However, if ##H## is a normal subgroup, then the left and right cosets are the same (##aH = Ha##), so we just call them cosets, and the set of cosets does form a group. In this case, the product ##aHbH## can be simplified as follows:
$$aHbH = a(Hb)H = a(bH)H = abHH = abH$$

Thank you for the good explanation.
 

What is a Cayley table?

A Cayley table is a visual representation of the group operation for a given group. It lists all of the possible combinations of elements in the group and the result of applying the group operation to those combinations.

What is the purpose of a Cayley table?

The purpose of a Cayley table is to show the structure and properties of a given group. It helps to identify the group's elements, operations, and relationships between elements.

How do you read a Cayley table?

In a Cayley table, the rows and columns represent the elements of the group, and the cells represent the result of applying the group operation to those elements. The first row and column usually list the identity element, and the cells in the first row and column should match the element in the corresponding row or column. To find the result of applying the group operation to two elements, locate the row of the first element and the column of the second element, and the cell where the row and column intersect will show the result.

What are some properties of a Cayley table?

A Cayley table is always square, meaning it has the same number of rows and columns. It is also symmetric across the main diagonal, meaning the elements above and below the diagonal are reflections of each other. Additionally, each element in the Cayley table appears exactly once in each row and column.

Can a Cayley table show all the properties of a group?

No, a Cayley table cannot show all the properties of a group. It only shows the group operation and the relationships between elements. Other properties, such as the identity element, inverse elements, and closure, may not be readily apparent from a Cayley table and may require further analysis to determine.

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