Group Table: Understanding the Identity Element and Avoiding Repeated Values

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In summary, the conversation discusses filling in a group table and determining the identity element. It is noted that there cannot be repeated values in a row or column, and the table should include all four group elements in each row and column. The conversation concludes with a suggested table that has the identity element on the main diagonal.
  • #1
Homework Statement
Group table
Relevant Equations
n/a
*stuv
ss?v?uv?
ttv
uu?
vv?

Since s*u=u does that mean s is the identity element? Then I know there can't be repeated values in a row or column so I need to us that to somehow fill in the rest of the blank spaces?
 

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  • #2
joelkato1605 said:
Homework Statement:: Group table
Relevant Equations:: n/a

*stuv
ss?v?uv?
ttv
uu?
vv?

Since s*u=u does that mean s is the identity element?
Yes, but then s*t=s*v=v cannot be!
Then I know there can't be repeated values in a row or column ...
Right.
... so I need to us that to somehow fill in the rest of the blank spaces?
No. You have to fill them with an element, since every multiplication has to be defined, and blank is no group element.

Hint: There are two possible solutions.
 
  • #3
Right, do I need to use something like U*T=S*U*T in some way?
 
  • #4
You have to decide, which element is the identity. Seems, that ##s## has this role. So ##s\cdot a= a## for any other group element. This gives you the first row and first column.

Before you check associativity, look whether you can fill up the remaining ##9## places according to the rule: all ##4## elements must occur in each row and each column! Do you know why there is such a rule?
 
  • #5
So the table should look like this
stuv
sstuv
ttvsu
uusvt
vvuts
 
  • #7
joelkato1605 said:
So the table should look like this
stuv
sstuv
ttvsu
uusvt
vvuts
What do you get if you put all ##s## on the main diagonal?
 

1. What is a group table?

A group table is a visual representation of a group's operation, where the elements of the group are listed in rows and columns. The table shows how each element combines with another element in the group to produce a third element.

2. What is the identity element in a group table?

The identity element in a group table is the element that, when combined with any other element in the group, results in that same element. In other words, it is the element that has no effect on the other elements in the group when the group's operation is applied.

3. Why is it important to understand the identity element in a group table?

Understanding the identity element is important because it helps us to identify the structure and properties of a group. It also allows us to solve equations and perform calculations within the group more efficiently.

4. How can we avoid repeated values in a group table?

To avoid repeated values in a group table, we must ensure that each row and column contains a unique set of elements. This can be achieved by using a specific operation, such as multiplication or addition, and following certain rules, such as the commutative and associative properties.

5. What are some real-world applications of group tables?

Group tables have various applications in fields such as mathematics, computer science, and physics. They are used to study symmetry and patterns, analyze data, and design algorithms. In chemistry, group tables are used to understand the properties and behavior of elements in the periodic table. In coding theory, they are used to design error-correcting codes. In physics, group tables are used to study the symmetries of physical systems and particles.

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