Problem 1 Let $G$ be the symmetric group... [FREE SOLUTION]

Chapter 10: Problem 1

Let $G$ be the symmetric group $S(n)$ and $V$ be a complex vector space, with basis $\left\\{v_{1}, v_{2}, \ldots, v_{n}\right\\}$. For $\pi \in G$ and any $v=\lambda_{1} v_{1}+\cdots+\lambda_{n} v_{n}$ of $V$, define $$ \pi \cdot v=\lambda_{1} v_{\pi(1)}+\cdots+\lambda_{n} v_{\pi(n)} $$ Show that $V$ is a $G$-set, and find both $\operatorname{orb}(v)$ and $G_{v}$ when (a) $n=4$ and $v=v_{1}+v_{2}+v_{3}+v_{4} ;$ (b) $n=4$ and $v=v_{1}+v_{3}$.

Short Answer

Expert verified

(a) Orbit: $\{v_1 + v_2 + v_3 + v_4\}$, Stabilizer: $S(4)$. (b) Orbit: 6 vectors, Stabilizer: $\mathbb{Z}_2 \times \mathbb{Z}_2$.

Step by step solution

Define Action and Verify G-set

In order to show that $V$ is a $G$-set, we need to verify that the action $\pi \cdot v$ given in the problem is a function from $G \times V$ to $V$ and satisfies group action properties. For any $\pi, \sigma \in G$ and $v \in V$, the action is defined by $\pi \cdot v = \lambda_1 v_{\pi(1)} + \cdots + \lambda_n v_{\pi(n)}$. The two properties we need to check are: 1. Identity: $e \cdot v = v$ for an identity permutation $e$.2. Compatibility: $(\pi \sigma) \cdot v = \pi \cdot (\sigma \cdot v)$.Both properties hold for the given action, confirming that $V$ is a $G$-set.

Calculate Orbit (a)

To find $\operatorname{orb}(v)$ with $v = v_1 + v_2 + v_3 + v_4$ when $n = 4$, apply all possible permutations $\pi \in S(4)$ on $v$. Each permutation $\pi$ rearranges the coefficients $\lambda_1, \lambda_2, \lambda_3, \lambda_4$, but since each term is present once, all permutations result in the same vector $v$. Therefore, $\operatorname{orb}(v) = \{v_1 + v_2 + v_3 + v_4\}$ which is a single-element set.

Calculate Stabilizer (a)

The stabilizer $G_v$, or the set of all group elements that leave $v$ unchanged, for $v = v_1 + v_2 + v_3 + v_4$ includes all permutations $\pi$ that do not change the vector. Since all permutations leave $v$ unchanged (because every $v_i$ occurs with coefficient 1), $G_v = S(4)$.

Calculate Orbit (b)

For the vector $v = v_1 + v_3$ with $n = 4$, the permutations will shuffle the positions of 1, 0, and turn the non-included spots into 0. The orbit consists of all vectors obtainable by permuting indices 1 and 3 among the four available positions: - $v_1 + v_3$- $v_1 + v_2$- $v_1 + v_4$- $v_2 + v_3$- $v_2 + v_4$- $v_3 + v_4$.Thus, $\operatorname{orb}(v) = \{v_1 + v_3, v_1 + v_2, v_1 + v_4, v_2 + v_3, v_2 + v_4, v_3 + v_4\}$.

Calculate Stabilizer (b)

The stabilizer $G_v$ for $v = v_1 + v_3$ are the permutations that switch $v_1$ and $v_3$ with each other without altering the coefficient locations. This includes transpositions within the set $\{1, 3\}$ and the identity permutation: $\{e, (1\;3), (2\;4), (1\;3)(2\;4)\}$ which form a subgroup isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_2$.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Symmetric Group

The symmetric group, denoted as $ S(n) $, is an essential concept in group theory that is made up of all possible permutations of $ n $ elements. These permutations represent all the ways that $ n $ different objects can be arranged or ordered.

Example: For $ n = 3 $, the symmetric group $ S(3) $ consists of the permutations:$ \{e, (12), (13), (23), (123),(132)\} $.
Identity Element: Represented as $ e $, the identity does nothing and leaves every object in its original position.

The symmetric group $ S(n) $ is significant because it is a non-Abelian group for $ n > 2 $, meaning the order of permutations matters. This property allows for complex symmetries and is foundational in studying more advanced algebraic structures.

Vector Spaces

A vector space is a collection of vectors along with the rules of vector addition and scalar multiplication. These spaces form the backbone of linear algebra and are characterized by a few essential properties:

Addition: Can add two vectors to get a new vector.
Scalar Multiplication: Can scale vectors by numbers from a field like the real or complex numbers.
Zero Vector: Contains a zero vector which, when added to any vector, does not change it.

In our exercise, $ V $ is a complex vector space with basis vectors $ \{v_1, v_2, \ldots, v_n\} $. Each vector in $ V $ can be expressed as a linear combination of these basis vectors. The roles these vector spaces play in group action are crucial as they provide the canvas on which the symmetric group operates.

Group Action

Group action is a way to describe symmetries of objects using groups. When a group $ G $ acts on a set $ X $, it rearranges or transforms the elements in $ X $ via the elements of $ G $. A group action must satisfy two properties:

Identity: The identity element of $ G $ leaves every element in $ X $ unchanged.
Compatibility: The action is compatible with the group operation, meaning that for any group elements $ g, h \in G $, applying $ g $ and then $ h $ is the same as applying $ gh $.

In our problem, the symmetric group $ S(n) $ acts on the vector space $ V $ by permuting the coefficients of vectors according to its elements. This transforms each vector in a systematic way, confirming that $ V $ is indeed a $ G $-set.

Orbit

In the context of group action, an orbit is the set of all elements that a particular element can be transformed into by the action of a group. For a vector $ v $ in a vector space subjected to a group action, its orbit under a group $ G $ is $ \operatorname{orb}(v) = \{ g \cdot v \mid g \in G \} $.

Single Element Orbit: In the provided exercise, if each permutation brings us back to the same vector (like for $ v = v_1 + v_2 + v_3 + v_4 $), the orbit consists of a single element.
Multiple Element Orbit: Sometimes, the orbit includes multiple distinct vectors, as shown with $ v = v_1 + v_3 $, where permutations create multiple different combinations.

This concept helps to understand the reach of a group's action over an element within the set.

Stabilizer

A stabilizer in group action refers to the subset of the group that fixes an element, leaving it unchanged. This means that the stabilizer $ G_v $ of an element $ v $ is the set $ \{ g \in G \mid g \cdot v = v \} $. In simpler terms, these are group elements that do not alter the given element when the group action is applied.

Full Group as Stabilizer: When every permutation keeps the vector unchanged, as in the case for $ v = v_1 + v_2 + v_3 + v_4 $, the stabilizer is the entire group $ S(n) $.
Subgroup as Stabilizer: When some, but not all, transformations leave a vector unchanged (like $ v = v_1 + v_3 $), the stabilizer is just a subgroup of $ G $.

Understanding stabilizers is key to analyzing the symmetry properties of the objects under consideration.

91影视

Short Answer

Step by step solution

Define Action and Verify G-set

Calculate Orbit (a)

Calculate Stabilizer (a)

Calculate Orbit (b)

Calculate Stabilizer (b)

Key Concepts

Symmetric Group

Vector Spaces

Group Action

Orbit

Stabilizer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Mechanics Maths

Pure Maths

Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.