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Chapter 16: Problem 11

Find all subgroups in each of the following groups. a) \(\left(\mathbf{Z}_{12},+\right)\) b) \(\left(\mathbf{Z}_{11}^{*}, \cdot\right)\) c) \(S_{3}\)

Short Answer

Expert verified
The subgroups of \(\mathbf{Z}_{12}\) are \({\{0\}}, {\{0, 6\}}, {\{0, 4, 8\}}, {\{0, 3, 6, 9\}}, {\{0, 2, 4, 6, 8,10\}}, {\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}}\). The subgroups of \(\mathbf{Z}_{11}^{*}\) are \({\{1\}}, {\{1, 10\}}, {\{1, 3, 4, 5, 9\}}, {\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}}\). The subgroups of \(S_{3}\) are \({\{\epsilon\}}, {\{\epsilon, (12)\}}, {\{\epsilon, (13)\}}, {\{\epsilon, (23)\}}, {\{\epsilon, (123), (132)\}}, {\{\epsilon, (12), (13), (132)\}}, {\{\epsilon, (12), (23), (123)\}}, {\epsilon, (13), (23), (123), (132), (12), (132)\}}\).

Step by step solution

01

Determine Subgroups of \(\mathbf{Z}_{12}\)

The group \(\mathbf{Z}_{12}\) is the set of integers under addition modulo 12. The subgroups of a cyclic group are all cyclic themselves and are determined by the divisors of the order of the group. The divisors of 12 are 1, 2, 3, 4, 6, and 12. So, subgroups exist corresponding to each of these divisors. They are \({\{0\}}, {\{0, 6\}}, {\{0, 4, 8\}}, {\{0, 3, 6, 9\}}, {\{0, 2, 4, 6, 8,10\}}, {\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}}\) respectively.
02

Determine Subgroups of \(\mathbf{Z}_{11}^{*}\)

\(\mathbf{Z}_{11}^{*}\) is the set of integers under multiplication modulo 11. This group has order 10 making it cyclic. The divisors of 10 are 1, 2, 5, 10. Equivalent to previous step, the subgroups are \({\{1\}}, {\{1, 10\}}, {\{1, 3, 4, 5, 9\}}, {\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}}\) respectively.
03

Determine Subgroups of \(S_{3}\)

\(S_{3}\) is the set of all permutations of three elements. \(S_{3}\) has 6 permutations (elements): \(\epsilon, (12), (13), (23), (123), (132)\). The subgroups of \(S_{3}\) are given by: \({\{\epsilon\}}, {\{\epsilon, (12)\}}, {\{\epsilon, (13)\}}, {\{\epsilon, (23)\}}, {\{\epsilon, (123), (132)\}}, {\{\epsilon, (12), (13), (132)\}}, {\{\epsilon, (12), (23), (123)\}}, {\epsilon, (13), (23), (123), (132), (12), (132)\}}\) respectively, where \(\epsilon\) represents the identity permutation.

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Most popular questions from this chapter

If \(\gamma=\left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3\end{array}\right) \in S_{4}\), how many cosets does \(\langle\gamma\rangle\) determine?

In \(S_{5}\) find an element of order \(n\), for all \(2 \leq n \leq 5\). Also determine the (cyclic) subgroup of \(S_{5}\) that each of these elements generates.

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Let $$ H=\left[\begin{array}{lllllll} 1 & 1 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 & 1 \end{array}\right] $$ be the parity-check matrix for a Hamming \((7,4)\) code. a) Encode the following messages: \(\begin{array}{llllll}1000 & 1100 & 1011 & 1110 & 1001 & 1111 .\end{array}\) b) Decode the following received words: \(\begin{array}{cccc}1100001 & 1110111 & 0010001 & 0011100\end{array}\) c) Construct a decoding table consisting of the syndromes and coset leaders for this code. d) Use the result in part (c) to decode the received words given in part (b).

a) Find all the elements of order 10 in \(\left(\mathbf{Z}_{40},+\right)\). b) Let \(G=\langle a\rangle\) be a cyclic group of order 40 . Which elements of \(G\) have order \(10 ?\)
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