/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Let \(G=S_{4}\). (a) For \(\alph... [FREE SOLUTION] | 91影视

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Chapter 16: Problem 1

Let \(G=S_{4}\). (a) For \(\alpha=\left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1\end{array}\right)\), find the subgroup \(H=\langle\alpha\rangle\). (b) Determine the left cosets of \(H\) in \(G\).

Short Answer

Expert verified
The subgroup \(H=\langle\alpha\rangle\) generated by \(\alpha\) is \{e, (1 2 3 4), (1 3 4 2), (1 4 2 3)\}, and there are 6 left cosets of \(H\) in \(G\) as \(|G|=24\) and \(|H|=4\).

Step by step solution

01

Understanding the permutation \(\alpha\)

The permutation \(\alpha\) is given in two-row notation. The top row represents the elements being permuted, and the bottom row represents the targets of these elements under the permutation. So \(\alpha\) consists of the cycle (1 2 3 4)
02

Finding the subgroup generated by \(\alpha\)

To find the subgroup \(H=\langle\alpha\rangle\), generate all the elements by repeatedly applying the operation of \(\alpha\). Start with the identity permutation \(e\), then apply \(\alpha\) to get (1 2 3 4), apply \(\alpha\) again to get (1 3 4 2), apply \(\alpha\) once more to get (1 4 2 3), and apply \(\alpha\) one more time to return to the identity. Thus, \(H=\langle\alpha\rangle=\{e, \alpha, \alpha^2, \alpha^3\}\
03

Generating the left cosets of \(H\) in \(G\)

Generating a left coset of \(H\) in \(G\) involves operating each element in \(G\) on \(H\). As \(G=S_4\) has 24 elements (4!), ideally this procedure should be repeated 24 times. However, noticing that \(|H|=4\), a property of cosets ensures that all left cosets have the same order as \(H\). So, there only can be 24/4=6 different left cosets of \(H\) in \(G\).

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Most popular questions from this chapter

Given \(n \in \mathbf{Z}^{+}\), let the set \(M(n, k) \subseteq \mathbf{Z}_{2}^{n}\) contain the maximum number of code words of length \(n\), where the minimum distance between code words is \(2 k+1\). Prove that $$ \frac{2^{n}}{\sum_{l=0}^{2 k}\left(\begin{array}{l} n \\ l \end{array}\right)} \leq|M(n, k)| \leq \frac{2^{n}}{\sum_{l=0}^{k}\left(\begin{array}{l} n \\ \imath \end{array}\right)} $$ (The upper bound on \(|M(n, k)|\) is called the Hamming bound; the lower bound is referred to as the Gilbert bound.)

For \(n \geq 1\), if \(\sigma, \tau \in S_{n}\), define the distance \(d(\sigma, \tau)\) between \(\sigma\) and \(\tau\) by $$ d(\sigma, \tau)=\max \\{\mid \sigma(i)-\tau(i) \| 1 \leq i \leq n\\} $$ a) Prove that the following properties hold for \(d\). i) \(d(\sigma, \tau) \geq 0\) for all \(\sigma, \tau \in S_{n}\) ii) \(d(\sigma, \tau)=0\) if and only if \(\sigma=\tau\) iii) \(d(\sigma, \tau)=d(\tau, \sigma)\) for all \(\sigma, \tau \in S_{n}\) iv) \(d(\rho, \tau) \leq d(\rho, \sigma)+d(\sigma, \tau)\), for all \(\rho, \sigma, \tau \in S_{n}\) b) Let \(\epsilon\) denote the identity element of \(S_{n}\) (that is, \(\epsilon(i)=i\) for all \(1 \leq i \leq n)\). If \(\pi \in S_{n}\) and \(d(\pi, \epsilon) \leq 1\), what can we say about \(\pi(n) ?\) c) For \(n \geq 1\) let \(a_{n}\) count the number of permutations \(\pi\) in \(S_{n}\), where \(d(\pi, \epsilon) \leq 1\). Find and solve a recurrence relation for \(a_{n}\).

a) In how many distinct ways can we 3-color the edges of a square that is free to move in three dimensions? b) In how many distinct ways can we 3-color both the vertices and the edges of such a square? c) For a square that can move in three dimensions, let \(k\), \(m\), and \(n\) denote the number of distinct ways in which we can 3-color its vertices (alone), its edges (alone), and both its vertices and edges, respectively. Does \(n=k m\) ? (Give a geometric explanation.)

a) Consider the group \(\left(\mathbf{Z}_{2} \times \mathbf{Z}_{2}, \oplus\right)\) where, for \(a, b, c, d \in\) \(\mathbf{Z}_{2},(a, b) \oplus(c, d)=(a+c, b+d)-\) the sums \(a+c\) and \(b+d\) are computed using addition modulo 2 . What is the value of \((1,0) \oplus(0,1) \oplus(1,1)\) in this group? b) Now consider the group \(\left(\mathbf{Z}_{2} \times \mathbf{Z}_{2} \times \mathbf{Z}_{2}, \oplus\right)\) where \((a, b, c) \oplus(d, e, f)=(a+d, b+e, c+f)\). (Here the sums \(a+d, b+e, c+f\) are computed using addition modulo 2.) What do we obtain when we add the seven nonzero (or nonidentity) elements of this group? c) State and prove a generalization that includes the results in parts (a) and (b).

Let \(f: G \rightarrow H\) be a group homomorphism onto \(H\). If \(G\) is abelian, prove that \(H\) is abelian.
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