Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 6
In \(S_{3}\) give an example of two elements \(x, y\) such that \((x \cdot y)^{2} \neq x^{2} \cdot y^{2}\).
Short Answer
Expert verified
In \(S_3\), \((12)\) and \((23)\) satisfy \((x \cdot y)^2 \neq x^2 \cdot y^2\).
Step by step solution
01
Understand the Symmetric Group S_3
The symmetric group \(S_3\) is the group of all permutations of the set \(\{1, 2, 3\}\) and has 6 elements. These elements can be expressed in cycle notation: \(()\), \((12)\), \((13)\), \((23)\), \((123)\), and \((132)\). We are asked to find two elements \(x\) and \(y\) such that \((x \cdot y)^2 eq x^2 \cdot y^2\).
02
Choose Elements x and y
Select \(x = (12)\) and \(y = (23)\). These elements are transpositions, and in symmetric groups, it is known that transpositions do not commute. This property will help us in the next calculations.
03
Calculate x \cdot y and (x \cdot y)^2
First, compute the product \(x \cdot y = (12)(23)\). This composition results in the cycle \((123)\). Next, calculate \((x \cdot y)^2 = ((123))^2 = (132)\), because cycling \((123)\) twice gives us \((132)\).
04
Calculate x^2 and y^2, then x^2 \cdot y^2
Both \(x\) and \(y\) are transpositions, so \(x^2 = (12)(12) = ()\) and \(y^2 = (23)(23) = ()\). Therefore, \(x^2 \cdot y^2 = () \cdot () = ()\), which is the identity permutation.
05
Compare (x \cdot y)^2 with x^2 \cdot y^2
Now that we have \((x \cdot y)^2 = (132)\) and \(x^2 \cdot y^2 = ()\), we see that \((132) eq ()\). This satisfies the condition \((x \cdot y)^2 eq x^2 \cdot y^2\).
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.
Group Theory
Group theory is a fascinating area within abstract algebra that studies algebraic structures called groups. Groups essentially provide a formal framework for analyzing symmetry and are fundamental in various fields of mathematics and science. A group is a set equipped with an operation that combines any two elements to form a third element, ensuring four main properties:
- Closure: The operation must produce a result that is also within the group.
- Associativity: Changing the grouping of operations does not affect the outcome, meaning \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
- Identity Element: There must be an element that, when combined with any element in the group, leaves the other element unchanged, like a neutral player.
- Inverse Element: For each element in the group, there must be another element that reverses the effect of combining them, leading back to the identity.
Permutations
Permutations are arrangements or rearrangements of a set of objects. When we think of permutations in the context of group theory, we are looking at the possible rearrangements of elements within a particular set.
In the symmetric group \(S_3\), permutations include every possible way to reorder the set \(\{1, 2, 3\}\).
The elements of \(S_3\) are:
In the symmetric group \(S_3\), permutations include every possible way to reorder the set \(\{1, 2, 3\}\).
The elements of \(S_3\) are:
- The identity permutation \(()\), which doesn't change anything.
- Two-element swaps (transpositions) like \((12)\), \((13)\), and \((23)\).
- Cyclic permutations \((123)\) and \((132)\), which rotate all elements among the three positions.
Cycle Notation
Cycle notation provides a compact way of representing permutations by denoting the specific cycles in which elements are permuted. This notation is especially useful when dealing with permutations in symmetric groups.
In cycle notation, a cycle \((a_1 \, a_2 \, ... \, a_k)\) indicates that element \(a_1\) goes to \(a_2\), \(a_2\) goes to \(a_3\), and so on, with \(a_k\) ultimately mapping back to \(a_1\).
For example, in \(S_3\), the cycle \((123)\) means:
In cycle notation, a cycle \((a_1 \, a_2 \, ... \, a_k)\) indicates that element \(a_1\) goes to \(a_2\), \(a_2\) goes to \(a_3\), and so on, with \(a_k\) ultimately mapping back to \(a_1\).
For example, in \(S_3\), the cycle \((123)\) means:
- 1 moves to position of 2
- 2 moves to the position of 3
- 3 moves to the position of 1
Transpositions
Transpositions are specific permutations that swap only two elements in a set and are fundamental components in permutations. In a symmetric group like \(S_3\), transpositions include \((12)\), \((13)\), and \((23)\), each swapping their respective pair of elements while leaving the third unchanged.
The importance of transpositions goes beyond their simplicity:
The importance of transpositions goes beyond their simplicity:
- They are the building blocks of all permutations in a symmetric group. This means any permutation within the symmetric group can be expressed as a product of transpositions.
- They exhibit the nature of non-commutativity in permutations. For example, \((12)(23)\) is not the same as \((23)(12)\), highlighting that the order of operations matters.
- Transpositions are their own inverses, meaning when you square a transposition, you get the identity element. For instance, \((12)^2 = ()\).
