Chapter 29, Symmetry and Counting Video Solutions, Contemporary Abstract Algebra

Problem 1

Determine the number of ways in which the four corners of a square can be colored with two colors. (It is permissible to use a single color on all four corners.)

Nicole Smina

Numerade Educator

00:52

Problem 2

Determine the number of different necklaces that can be made using 13 white beads and 3 black beads.

Hossam Mohamed

Numerade Educator

01:35

Problem 3

Determine the number of ways in which the vertices of an equilateral triangle can be colored with five colors so that at least two colors are used.

Christopher Stanley

Numerade Educator

01:41

A benzene molecule can be modeled as six carbon atoms arranged in a regular hexagon in a plane. At each carbon atom, one of three radicals $\mathrm{NH}_{2}, \mathrm{COOH}$, or $\mathrm{OH}$ can be attached. How many such compounds are possible? (Make no distinction between single and double bonds between the atoms.)

Benjamin Angeles

Numerade Educator

02:08

Problem 5

Suppose that in Exercise 4 we permit only $\mathrm{NH}_{2}$ and $\mathrm{COOH}$ for the radicals. How many compounds are possible?

Patha Sharma

Numerade Educator

01:07

Problem 6

Determine the number of ways in which the faces of a regular dodecahedron (regular 12 -sided solid) can be colored with three colors.

Kratika Bhadauria

Numerade Educator

01:18

Problem 7

Determine the number of ways in which the edges of a square can be colored with six colors so that no color is used on more than one edge.

Nicole Smina

Numerade Educator

01:18

Problem 8

Determine the number of ways in which the edges of a square can be colored with six colors with no restriction placed on the number of times a color can be used.

Nicole Smina

Numerade Educator

01:26

Problem 9

Determine the number of different 11 -bead necklaces that can be made using two colors.

Vysakh M

Numerade Educator

00:28

Problem 10

Determine the number of ways in which the faces of a cube can be colored with three colors.

Sarah Wharton

Numerade Educator

01:35

Problem 11

Suppose a cake is cut into six identical pieces. How many ways can we color the cake with $n$ colors assuming that each piece receives one color?

Christopher Stanley

Numerade Educator

00:51

Problem 12

How many ways can the five points of a five-pointed crown be painted if three colors of paint are available?

Hast Aggarwal

Numerade Educator

01:58

Problem 13

Let $G$ be a finite group and let $\operatorname{sym}(G)$ be the group of all permutations on $G$. For each $g$ in $G$, let $\phi_{e}$ denote the element of $\operatorname{sym}(G)$ defined by $\phi_{g}(x)=g x g^{-1}$ for all $x$ in $G$. Show that $G$ acts on itself under the action $g \rightarrow \phi_{g^{\cdot}}$ Give an example in which the mapping $g \rightarrow \phi_{g}$ is not oneto-one.

Varsha Aggarwal

Numerade Educator

01:40

Problem 14

Let $G$ be a finite group, let $H$ be a subgroup of $G$, and let $S$ be the set of left cosets of $H$ in $G$. For each $g$ in $G$, let $\gamma_{g}$ denote the element of $\operatorname{sym}(S)$ defined by $\gamma_{g}(x H)=g x H$. Show that $G$ acts on $S$ under the action $g \rightarrow \gamma_{p^{*}}$

Varsha Aggarwal

Numerade Educator

03:44

Problem 15

For a fixed square, let $L_{1}$ be the perpendicular bisector of the top and bottom of the square and let $L_{2}$ be the perpendicular bisector of the left and right sides. Show that $D_{4}$ acts on $\left\{L_{1}, L_{2}\right\}$ and determine the kernel of the mapping $g \rightarrow \gamma_{g}$.

Ashley Volpe

Numerade Educator

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15