• Home
  • Textbooks
  • A Book of Abstract Algebra
  • PERMUTATIONS OF A FINITE SET

A Book of Abstract Algebra

Charles C. Pinter

Chapter 8

PERMUTATIONS OF A FINITE SET - all with Video Answers

Educators


Section 1

A

01:04

Problem 1

Compute each of the following products in $S_{9}$. (Write your answer as a single permutation.).
(a) $(145)(37)(682)$
(b) $(17)(628)(9354)$
(c) $(71825)(36)(49)$
(d) $(12)(347)$
(e) $(147)(1678)(74132)$
$(f)(6148)(2345)(12493)$

Narayan Hari
Narayan Hari
Numerade Educator
01:47

Problem 2

Write each of the following permutations in $S_{9}$ as a product of disjoint cycles:
(a) $\left(\begin{array}{lllllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 4 & 9 & 2 & 5 & 1 & 7 & 6 & 8 & 3\end{array}\right)$
(b) $\left(\begin{array}{lllllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 7 & 4 & 9 & 2 & 3 & 8 & 1 & 6 & 5\end{array}\right)$
(c) $\left(\begin{array}{lllllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 7 & 9 & 5 & 3 & 1 & 2 & 4 & 8 & 6\end{array}\right)$
(d) $\left(\begin{array}{ccccccccc}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 9 & 8 & 7 & 4 & 3 & 6 & 5 & 1 & 2\end{array}\right)$

Taha T
Taha T
Numerade Educator
01:50

Problem 3

Express each of the following as a product of transpositions in $S_{8}$.
(a) $(137428)$
(b) $(416)(8235)$
(c) $(123)(456)(1574)$
(d) $\pi=\left(\begin{array}{llllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 3 & 1 & 4 & 2 & 8 & 7 & 6 & 5\end{array}\right)$

Allison Knapp
Allison Knapp
Numerade Educator
02:13

Problem 4

If $\alpha=(3714), \beta=(123)$, and $y=(24135)$ in $S_{7}$, express each of the following as a product of disjoint cycles:
(a) $\alpha^{-1} \beta$
(b) $y^{-1} \alpha$
(c) $\alpha^{2} \beta$
$(d) \beta^{2} \alpha \gamma$
(e) $\gamma^{4}$
$(f) \gamma^{3} \alpha^{-1}$
(g) $\beta^{-1} \gamma$
$(h) \alpha^{-1} \gamma^{2} \alpha$

Hunza Gilgit
Hunza Gilgit
Numerade Educator
00:40

Problem 5

In $S_{5}$, write (12345) in five different ways as a cycle, and in five different ways as a product of transpositions.

Aymara Gallardo
Aymara Gallardo
Numerade Educator
01:48

Problem 6

In $S_{5}$, express each of the following as the square of a cycle (that is, express as $\alpha^{2}$ where $\alpha$ is a cycle):
(a) $(132)$
(b) $(12345)$
(c) $(13)(24)$

Srilakshmi E K
Srilakshmi E K
Numerade Educator