Find Order of Subgroup of 4x4 Matrices in G

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In summary, the conversation discusses finding the order of a subgroup in the multiplicative group G of 4x4 matrices, specifically the matrix A. The conversation includes trying to find A^3 and realizing it is the identity matrix, ultimately leading to the conclusion that the order of A is 3.
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duki
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Homework Statement



Thus far in my studying I've been able to at least have a sense of where to start solving the problems... until now.

Find the order of the subgroup of the multiplicative group G of 4x4 matrice generated by:

| 0 1 0 0 |
| 0 0 0 1 |
| 0 0 1 0 |
| 1 0 0 0 |

Recall the identity e:

| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |

Homework Equations



The Attempt at a Solution



No clue, whatsoever. :(
 
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  • #2
Call your matrix A. What's A3?
 
  • #3
I'm not sure? =(
 
  • #4
What do you mean, "I'm not sure"? Cube the matrix, i.e. multiply it by itself three times. Then you'll know what A^3 is. You aren't giving VKint the help deserved of the clue.
 
  • #5
Ok, had to remember how to multiply matrices. I got the identity, e.
 
  • #6
So did I. Problem solved, right?
 
  • #7
I don't quite understand. So is the order 3 because that's how many times I had to multiply it by itself to get the identity?
 
  • #8
A isn't the identity, A^2 isn't the identity, A^3 is. So yes, the order is 3. Look up 'order of a group element'.
 
  • #9
Awesome, thanks.
 

FAQ: Find Order of Subgroup of 4x4 Matrices in G

What is a subgroup?

A subgroup is a subset of a larger group that also forms a group under the same operation as the larger group. In other words, a subgroup is a smaller group that is contained within a larger group.

What is a 4x4 matrix?

A 4x4 matrix is a square matrix that has four rows and four columns. It is a mathematical object that is commonly used in linear algebra and can represent a variety of transformations, such as rotations and reflections.

How do you find the order of a subgroup of 4x4 matrices in G?

To find the order of a subgroup of 4x4 matrices in G, you first need to determine the elements of the subgroup. Then, you need to count the number of elements in the subgroup. This number will be the order of the subgroup.

What does the order of a subgroup represent?

The order of a subgroup represents the number of elements in the subgroup. It is an important concept in group theory as it can help determine the structure and properties of a group. It can also be used to find other subgroups and understand the relationships between them.

Why is finding the order of a subgroup of 4x4 matrices in G important?

Finding the order of a subgroup of 4x4 matrices in G is important for understanding the structure and properties of the larger group, G. It can also help in solving complex problems and determining the relationships between different subgroups. Additionally, the order of a subgroup can provide insights into the symmetry of a system or object represented by the 4x4 matrices.

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