Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 6: Problem 257
Explain why the set of all permutations of four elements is a permutation group. How many elements does this group have? This group is called the symmetric group on four letters and is denoted by \(S_{4}\).
Short Answer
Expert verified
The set of all permutations of four elements forms a group called \(S_{4}\) with 24 elements.
Step by step solution
01
- Understanding Permutations
A permutation of a set is a way of arranging its elements in a specific order. For four elements, say \(1, 2, 3, 4\), a permutation is any arrangement of these numbers, such as \(2, 1, 4, 3\).
02
- Defining a Group
A group is a set equipped with an operation that combines any two of its elements to form another element of the set, and must satisfy closure, associativity, have an identity element, and every element must have an inverse.
03
- Closure Property
The set of all permutations of four elements, denoted by \(S_{4}\), is closed under composition of permutations. This means that the composition of any two permutations in \(S_{4}\) is also a permutation in \(S_{4}\).
04
- Associativity
The composition of permutations is associative. For any three permutations \(a, b, c \) in \(S_{4}\), we have \( (a \circ b) \circ c = a \circ (b \circ c) \).
05
- Identity Element
The identity permutation, which leaves every element in its original position, serves as the identity element in \(S_{4}\). Composing any permutation with the identity permutation results in the original permutation.
06
- Inverses
Every permutation has an inverse permutation that, when composed with the original, yields the identity permutation. For any permutation \( p \) in \(S_{4}\), there exists an inverse permutation \( p^{-1} \) such that \( p \circ p^{-1} = \text{id} \).
07
- Number of Elements in \(S_{4}\)
The number of elements in \(S_{4}\) is equal to the number of permutations of four elements. This is calculated as \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).
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.
Permutations
A permutation is essentially an arrangement of elements in a particular order. Imagine having four elements labeled as 1, 2, 3, and 4. Any reordering of these elements, such as 2, 1, 4, 3, is considered a permutation. In total, there are 24 different ways to arrange four distinct elements, as the number of permutations of four elements is given by the factorial operation 4!, calculated as 4 脳 3 脳 2 脳 1 = 24. Permutations play a fundamental role in the symmetric group.
Group Theory
Group theory is a branch of mathematics that studies algebraic structures known as groups. In the context of permutations, a group is a set together with an operation (in this case, composition of permutations) that satisfies four key properties: closure, associativity, identity, and invertibility.
Understanding groups can help in various fields, including solving puzzles, coding theory, and even physics.
Groups can simplify problem-solving by offering a structured way to think about symmetry and operations performed on sets.
Understanding groups can help in various fields, including solving puzzles, coding theory, and even physics.
Groups can simplify problem-solving by offering a structured way to think about symmetry and operations performed on sets.
Associativity
Associativity is one of the core properties required for a set with a binary operation to be considered a group. In the symmetric group S4, the operation is the composition of permutations. This property means that the way we group operations does not affect the result. For three permutations a, b, and c, the equation (a 鈭� b) 鈭� c = a 鈭� (b 鈭� c) holds true. This ensures consistency when multiple permutations are composed together, allowing us to focus on finding solutions rather than worrying about the order in which we combine them.
Identity Element
The identity element is a special element in the group that acts as a 'do nothing' step in the context of the operation. For the symmetric group S4, the identity permutation leaves every element in its original position, such as 1, 2, 3, 4 remaining 1, 2, 3, 4.
When you compose any permutation with the identity permutation, it leaves the permutation unchanged, like a 鈭� id = a and id 鈭� a = a.
Without an identity element, the group structure would fail to hold, which makes the identity element crucial for group theory.
When you compose any permutation with the identity permutation, it leaves the permutation unchanged, like a 鈭� id = a and id 鈭� a = a.
Without an identity element, the group structure would fail to hold, which makes the identity element crucial for group theory.
Inverse Element
In a group, every element must have an inverse. For any permutation p in S4, there exists an inverse permutation p鈦宦�. When a permutation and its inverse are composed, they yield the identity permutation: p 鈭� p鈦宦� = id.
For example, if a permutation swaps 1 and 2, its inverse will also swap 1 and 2 to return everything to the original order.
This invertibility property ensures that we can 'undo' any permutation operation, which is essential for maintaining the group's structure and solving equations involving permutations.
For example, if a permutation swaps 1 and 2, its inverse will also swap 1 and 2 to return everything to the original order.
This invertibility property ensures that we can 'undo' any permutation operation, which is essential for maintaining the group's structure and solving equations involving permutations.
