Problem 15 Let \(H\) be a nontrivial normal... [FREE SOLUTION]

Chapter 5: Problem 15

Let \(H\) be a nontrivial normal subgroup of \(A_{n}, n \geq 6 .\) Show that there exists \(\tau \in H\) and \(1 \leq i \leq n\) such that \(\tau(i)=i\) and \(\tau\) is not the identity permutation.

Short Answer

Expert verified

In \(A_n\), a 3-cycle like \((123)\) is in \(H\) and has several fixed points.

Step by step solution

Understanding Normal Subgroups and Symmetric Properties

A subgroup is normal if it is invariant under conjugation, which means for every element \(h\) in \(H\) and \(g\) in \(A_n\), the element \(ghg^{-1}\) is also in \(H\). Since \(A_n\) is a simple group for \(n \geq 5\), the only normal subgroups are the trivial group and the group itself. Here, \(H\) is nontrivial and normal, hence \(H=A_n\) for \(n \geq 6\).

Analyzing the Properties of Permutations in \(A_n\)

Since \(H=A_n\), any permutation \(\tau\) in \(H\) corresponds to a permutation in \(A_n\) which is an even permutation. An even permutation is a permutation that can be expressed as a product of an even number of transpositions. We are looking for such a \(\tau\) with at least one fixed point \(i\) such that \(\tau(i)=i\).

Using the Degree of Permutations to Identify a Suitable \(\tau\)

In \(A_n\), every permutation fixes at least one point in a nontrivial cycle decomposition when \(n \geq 6\). Specifically, permutations like transpositions or 3-cycles will have fixed points since their cycle notation necessarily leaves elements unchanged. We need a permutation that is not the identity, hence cannot fix all points.

Concluding with an Example Permutation

Consider a simple 3-cycle \(\tau = (123) \) in \(A_n\) for \(n \geq 6\). This permutation fixes all elements except 1, 2, and 3, thus having fixed points such as \(\forall\ i \geq 4, \tau(i) = i\). Consequently, \(\tau\) satisfies the requirement of not being the identity while having fixed points.

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Key Concepts

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

alternating group

The alternating group, denoted as \( A_n \), is a critical concept in group theory. It is defined as the group of even permutations of a set of \( n \) elements. An even permutation is one that can be written as a product of an even number of transpositions, which are simple swaps of two elements. The alternating group is a subgroup of the symmetric group \( S_n \), which consists of all possible permutations.

Here are a few key properties of \( A_n \):

\( A_n \) is nontrivial for \( n \geq 3 \), meaning it contains permutations other than just the identity.
It is a normal subgroup of \( S_n \), which means it remains unchanged when conjugated by any element of \( S_n \).
For \( n \geq 5 \), \( A_n \) is a simple group. This implies it has no nontrivial, proper normal subgroups, making it a building block for more complex groups.

Understanding the alternating group helps in solving various algebra problems, including those involving cycle notations and fixed points.

permutations

Permutations are arrangements or rearrangements of elements of a set. Every permutation can be expressed in terms of cyclical motions known as cycle notation. Cycle notation simplifies the representation of permutations by grouping elements together in a cyclic manner, which dictates how elements are moved around.

Cyclic notation example: The permutation \( (123) \) means element 1 goes to position 2, 2 goes to position 3, and 3 goes to position 1.
Each cycle in the notation indicates a sequence of positions cycled among.
Permutations can be odd or even, determined by the number of transpositions required to achieve them. An even permutation, as found in the alternating group \( A_n \), uses an even number of transpositions.

Understanding permutations is crucial because they form the foundation of symmetric and alternating groups. Recognizing even and odd permutations helps to classify elements within these groups, essential for proving solutions involving real-world applications.

fixed point

A fixed point in the context of permutations is an element of a set that remains unchanged when a permutation is applied. In mathematical terms, if a permutation \( \tau \) holds a fixed point \( i \), then \( \tau(i) = i \). Fixed points help simplify the analysis of permutations since they identify elements whose positions do not change.

Fixed points are common in cycle decompositions, where elements not included in the cycle notation are typically fixed points.
In larger permutations, especially those in \( A_n \) with \( n \geq 6 \), fixed points help identify non-identity elements.
For example, a 3-cycle like \( (123) \) in \( A_n \) leaves most elements unchanged, highlighting fixed points, typically defined as elements not within the cycle.

Recognizing fixed points is important when analyzing permutations to ensure they meet specific conditions, such as not being the identity permutation while having elements that remain unchanged in the set.

91影视

Let \(H\) be a nontrivial normal subgroup of \(A_{n}, n \geq 6 .\) Show that there exists \(\tau \in H\) and \(1 \leq i \leq n\) such that \(\tau(i)=i\) and \(\tau\) is not the identity permutation.

Short Answer

Step by step solution

Understanding Normal Subgroups and Symmetric Properties

Analyzing the Properties of Permutations in \(A_n\)

Using the Degree of Permutations to Identify a Suitable \(\tau\)

Concluding with an Example Permutation

Key Concepts

alternating group

permutations

fixed point

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