A question about linear algebra

In summary, there are 6 abelian groups of order 32 and we need at least 5 values of s to differentiate them based on their annihilator values.
  • #1
Artusartos
247
0
Classify the abelian groups of order 32.
a) In each case give the annihilator of the group along with [tex]dim_{Z_2} \frac{ker \mu_{2^s}}{ker \mu_{2^{s-1}}}[/tex]


for s=1,...,5. Where [tex]\mu_k(x) = kx [/tex] for all k.

b) If you know the annihilator of each of these groups, how many values of s (beginning with s=1) are needed to tell them apart?

My answer:

The abelian groups of order 32:

[tex]Z_{32}[/tex]

[tex] Z_{16} \bigoplus Z_2 [/tex]

[tex] Z_8 \bigoplus Z_4[/tex]

[tex] Z_8 \bigoplus Z_2 \bigoplus Z_2 [/tex]

[tex] Z_4 \bigoplus Z_2 \bigoplus Z_2 \bigoplus Z_2 [/tex]

[tex] Z_2 \bigoplus Z_2 \bigoplus Z_2 \bigoplus Z_2 \bigoplus Z_2[/tex]


For part a, if we look at [tex]Z_{32}[/tex], we have

[tex]dim_{Z_2} \frac{ker \mu_{2^s}}{ker \mu_{2^{s-1}}}[/tex]

=[tex]dim_{Z_2} \frac{Z_\bar{16}}{Z_\bar{32}}[/tex]

I'm kind of stuck now...can anybody please give me a hint?

Thanks in advance
 
Physics news on Phys.org
  • #2
!For part a, we haveZ_{32}: Annihilator = 0, dim_{Z_2} \frac{ker \mu_{2^s}}{ker \mu_{2^{s-1}}} = 0Z_{16} \bigoplus Z_2: Annihilator = 2, dim_{Z_2}\frac{ker \mu_{2^s}}{ker \mu_{2^{s-1}}} = 0Z_8 \bigoplus Z_4: Annihilator = 4, dim_{Z_2}\frac{ker \mu_{2^s}}{ker \mu_{2^{s-1}}} = 1Z_8 \bigoplus Z_2 \bigoplus Z_2: Annihilator = 8, dim_{Z_2}\frac{ker \mu_{2^s}}{ker \mu_{2^{s-1}}} = 2Z_4 \bigoplus Z_2 \bigoplus Z_2 \bigoplus Z_2: Annihilator = 16, dim_{Z_2}\frac{ker \mu_{2^s}}{ker \mu_{2^{s-1}}} = 3Z_2 \bigoplus Z_2 \bigoplus Z_2 \bigoplus Z_2 \bigoplus Z_2: Annihilator = 32, dim_{Z_2}\frac{ker \mu_{2^s}}{ker \mu_{2^{s-1}}} = 4For part b, we need at least 5 values of s to tell the groups apart (s=1,2,3,4,5).
 

FAQ: A question about linear algebra

What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations, vectors, and matrices. It is used to solve problems involving systems of linear equations, transformations, and geometric interpretations.

What are the applications of linear algebra?

Linear algebra has a wide range of applications in various fields such as physics, engineering, computer science, economics, and statistics. It is used in data analysis, image and signal processing, optimization problems, and machine learning algorithms.

What are the basic concepts in linear algebra?

The basic concepts in linear algebra include vectors, matrices, systems of linear equations, vector spaces, linear transformations, eigenvalues and eigenvectors, and determinants. These concepts are essential for understanding more complex topics in linear algebra.

What is the difference between a vector and a matrix?

A vector is a one-dimensional array of numbers, while a matrix is a two-dimensional array of numbers. Vectors are used to represent quantities that have magnitude and direction, while matrices are used to represent linear transformations and systems of linear equations.

How is linear algebra used in machine learning?

Linear algebra is crucial in machine learning as it provides the mathematical foundation for many algorithms such as linear regression, principal component analysis, and support vector machines. It is used to manipulate and analyze large datasets, make predictions, and create mathematical models to solve problems in various industries.

Back
Top