/*! 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

If \(G\) is a group of even order, prove that there is an element \(a \in G\) with \(a \neq e\) and \(a=a^{-1}\).

Express each of the following elements of \(S_{7}\) as a product of disjoint cycles. $$ \begin{aligned} \alpha &=\left(\begin{array}{lllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 2 & 4 & 6 & 7 & 1 & 5 & 3 \end{array}\right) \\ \beta &=\left(\begin{array}{lllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 6 & 5 & 2 & 1 & 7 & 4 \end{array}\right) \\ \gamma &=\left(\begin{array}{lllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 2 & 3 & 1 & 7 & 5 & 4 & 6 \end{array}\right) \\ \delta &=\left(\begin{array}{lllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 4 & 2 & 7 & 1 & 3 & 6 & 5 \end{array}\right) \end{aligned} $$

Let \(H\) and \(K\) be subgroups of a group \(G\), where \(e\) is the identity of \(G\). a) Prove that if \(|H|=10\) and \(|K|=21\), then \(H \cap K=\\{e\\}\). b) If \(|H|=m\) and \(|K|=n\), with \(\operatorname{gcd}(m, n)=1\), prove that \(H \cap K=\\{e\\}\)

a) Construct a decoding table (with syndromes) for the group code given by the generator matrix $$ G=\left[\begin{array}{lllll} 1 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 1 \end{array}\right] $$ b) Use the table from part (a) to decode the following received words. \(\begin{array}{cccc}11110 & 11101 & 11011 & 10100 \\ 10011 & 10101 & 11111 & 01100\end{array}\) c) Does this code correct single errors in transmission?

The \((5 m, m)\) five-times repetition code has encoding function \(E: \mathbf{Z}_{2}^{m} \rightarrow \mathbf{Z}_{2}^{5 m}\), where \(E(w)=w w w w w .\) Decoding with \(D: \mathbf{Z}_{2}^{5 m} \rightarrow \mathbf{Z}_{2}^{m}\) is accomplished by the majority rule. (Here we are able to correct single and double errors made in transmission.) a) With \(p=0.05\), what is the probability for the transmission and correct decoding of the signal \(0 ?\) b) Answer part (a) for the message 110 in place of the signal \(0 .\) c) For \(m=2\), decode the received word $$ r=0111001001 $$ d) If \(m=2\), find three received words \(r\) where \(D(r)=00\). e) For \(m=2\) and \(D: \mathbf{Z}_{2}^{10} \rightarrow \mathbf{Z}_{2}^{2}\), what is \(\left|D^{-1}(w)\right|\) for each \(w \in \mathbf{Z}_{2}^{2} ?\)
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