Chapter 4

In Exercises 1 through 8 (a) find the stabilizer \(G_{a}\) for each \(a\) indicated, (b) find the orbit \(O_{a},(\mathrm{c})\) check the orbit-stabilizer relation \(\left|O_{a}\right|=\left[G: G_{a}\right],(\mathrm{d})\) determine whether the action is transitive, and (e) determine whether \(G\) acts faithfully on \(X\). $$ X=\\{1,2,3\\} ; G=S_{3} ; a=1,2,3 $$

In Exercises 1 through 11 find the number of essentially different ways in which we can do what is described. Form a tetrahedral die by marking the four faces of a regular tetrahedron with one, two, three, or four dots, each number of dots appearing on exactly one face.

Find the order of a Sylow \(p\) -subgroup of \(G\) for the indicated \(p\) and \(G\) : \(\begin{array}{ll}\text { (a) } p=2 & G=S_{5}\end{array}\) (b) \(p=2 \quad G=A_{4}\) (c) \(p=2\) \(G=D_{6}\) (d) \(p=3 \quad G=\) any group of order 270

Determine how many nonisomorphic groups there are of the indicated order, and give examples of such groups. Groups of order 9

Describe the conjugacy classes and the class equation of an Abelian group.

Let \(\sigma=(12)(345)(6789)\) in \(S_{9}\). (a) Write down two permutations \(\rho, \tau\) that are conjugates of \(\sigma\) in \(S_{9}\). (b) Find permutations \(\phi, \chi, \psi\) such that \(\rho=\phi \sigma \phi^{-1}, \tau=\chi \sigma \chi^{-1}, \tau=\psi \rho \psi^{-1}\).

In Exercises 1 through 11 find the number of essentially different ways in which we can do what is described. Color the edges of a square, if six colors are available and no color is to be used more than once.

Let \(P\) be a Sylow 2 -subgroup of a group \(G\) of order 20 . If \(P\) is not a normal subgroup of \(G,\) how many conjugates does \(P\) have in \(G\) ?

Determine how many nonisomorphic groups there are of the indicated order, and give examples of such groups. Groups of order 10

Let \(G_{1}\) and \(G_{2}\) be two groups. Show that in \(G_{1} \times G_{2}\), the elements \((a, b)\) and \((c, d)\) are conjugates if and only if \(a\) and \(c\) are conjugates in \(G_{1},\) and \(b\) and \(d\) are conjugates in \(G_{2}\).

(a) find the stabilizer \(G_{a}\) for each \(a\) indicated, (b) find the orbit \(O_{a},(\mathrm{c})\) check the orbit-stabilizer relation \(\left|O_{a}\right|=\left[G: G_{a}\right],(\mathrm{d})\) determine whether the action is transitive, and (e) determine whether \(G\) acts faithfully on \(X\). $$ X=\\{1,2,3,4\\} ; G=S_{4} ; a=1,3,4 $$

In Exercises 1 through 11 find the number of essentially different ways in which we can do what is described. Paint two faces of a regular tetrahedron red and the other two faces green.

Find all Sylow \(p\) -subgroups of \(G\) for the indicated \(p\) and \(G\), and show that they are conjugates. $$ p=3, G=S_{4} $$

Determine how many nonisomorphic groups there are of the indicated order, and give examples of such groups. Groups of order 14

In Exercises 3 through 7 (a) show how \(X\) may be regarded as a \(G\) -set in a natural way or, in other words, describe a natural group action of \(G\) on \(X ;\) (b) show that the action has the two properties required by the definition of an action; and (c) give a permutation representation of the action. \(G=\\{e, g\\},\) the cyclic group of order 2. \(X=\) the vertices of an equilateral triangle.

In Exercises 3 through 8 describe the conjugacy classes and write the class equations of the indicated groups. $$ \mathbb{Z}_{2} \times S_{3} $$

(a) find the stabilizer \(G_{a}\) for each \(a\) indicated, (b) find the orbit \(O_{a},(\mathrm{c})\) check the orbit-stabilizer relation \(\left|O_{a}\right|=\left[G: G_{a}\right],(\mathrm{d})\) determine whether the action is transitive, and (e) determine whether \(G\) acts faithfully on \(X\). $$ X=\\{1,2,3,4\\} ; G=A_{4} \subseteq S_{4} ; a=1,3,4 $$

In Exercises 1 through 11 find the number of essentially different ways in which we can do what is described. Paint two faces of a cube red, two other faces blue, and the two remaining faces green.

Find all Sylow \(p\) -subgroups of \(G\) for the indicated \(p\) and \(G\), and show that they are conjugates. $$ p=2, G=S_{4} $$

Determine how many nonisomorphic groups there are of the indicated order, and give examples of such groups. Groups of order 49

(a) show how \(X\) may be regarded as a \(G\) -set in a natural way or, in other words, describe a natural group action of \(G\) on \(X ;\) (b) show that the action has the two properties required by the definition of an action; and (c) give a permutation representation of the action. \(G=\left\\{e, g, g^{2}\right\\},\) the cyclic group of order \(3 . X=\) the vertices of an equilateral triangle.

In Exercises 3 through 8 describe the conjugacy classes and write the class equations of the indicated groups. $$ \mathbb{Z}_{2} \times D_{4} $$

(a) find the stabilizer \(G_{a}\) for each \(a\) indicated, (b) find the orbit \(O_{a},(\mathrm{c})\) check the orbit-stabilizer relation \(\left|O_{a}\right|=\left[G: G_{a}\right],(\mathrm{d})\) determine whether the action is transitive, and (e) determine whether \(G\) acts faithfully on \(X\). \(X=\\{1,2,3,4,5,6,7,8\\} ;\) \(G=\left\\{\rho_{0}=\right.\) identity \(\left.,(1234)(57),(13)(24),(1432)(57)\right\\} \subseteq S_{7} ; a=1,3,6,7\)

In Exercises 1 through 11 find the number of essentially different ways in which we can do what is described. Color the six faces of a cube with six different colors, if seven colors are available and no color is to be used more than once.

Find all Sylow \(p\) -subgroups of \(G\) for the indicated \(p\) and \(G\), and show that they are conjugates. $$ p=2, G=A_{5} $$

Determine how many nonisomorphic groups there are of the indicated order, and give examples of such groups. Groups of order 99

(a) show how \(X\) may be regarded as a \(G\) -set in a natural way or, in other words, describe a natural group action of \(G\) on \(X ;\) (b) show that the action has the two properties required by the definition of an action; and (c) give a permutation representation of the action. \(G \approx \mathbb{Z}_{2} \times \mathbb{Z}_{2,}\) a noncyclic group of order \(4 . X=\) the vertices of a square.

In Exercises 3 through 8 describe the conjugacy classes and write the class equations of the indicated groups. $$ \mathrm{Z}_{3} \times S_{3} $$

Find the center \(Z\left(S_{3}\right)\) of \(S_{3}\).

(a) find the stabilizer \(G_{a}\) for each \(a\) indicated, (b) find the orbit \(O_{a},(\mathrm{c})\) check the orbit-stabilizer relation \(\left|O_{a}\right|=\left[G: G_{a}\right],(\mathrm{d})\) determine whether the action is transitive, and (e) determine whether \(G\) acts faithfully on \(X\). \(X=\) the four vertices of a square \\{1,2,3,4\\}\(; G=D_{4} ; a=1,2,3\)

In Exercises 1 through 11 find the number of essentially different ways in which we can do what is described. String three black and six white beads in a necklace, assuming the necklace can be turned over as well as rotated, and that beads of the same color are indistinguishable.

Find all Sylow 3 -subgroups and Sylow 5 -subgroups in \(S_{5}\).

Show that no group of the indicated order is simple. Groups of order 42

(a) show how \(X\) may be regarded as a \(G\) -set in a natural way or, in other words, describe a natural group action of \(G\) on \(X ;\) (b) show that the action has the two properties required by the definition of an action; and (c) give a permutation representation of the action. \(G \approx \mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2}, X=\) the vertices of a regular hexagon.

In Exercises 3 through 8 describe the conjugacy classes and write the class equations of the indicated groups. $$ S_{3} \times S_{3} $$

Find the center \(Z\left(S_{n}\right)\) of \(S_{n}\) for \(n>3\).

(a) find the stabilizer \(G_{a}\) for each \(a\) indicated, (b) find the orbit \(O_{a},(\mathrm{c})\) check the orbit-stabilizer relation \(\left|O_{a}\right|=\left[G: G_{a}\right],(\mathrm{d})\) determine whether the action is transitive, and (e) determine whether \(G\) acts faithfully on \(X\). \(X=\\{1,2,3,4\\},\) the four vertices of a square; \(G=\langle\rho\rangle,\) the subgroup gencrated by the rotation \(\rho\) of \(90^{\circ}\) in \(D_{4} ; a=1,3\)

Show that the intersection of all the Sylow 2 -subgroups in \(S_{4}\) is a normal ubgroup of \(S_{4}\) isomorphic to the Klein 4 -group.

Show that no group of the indicated order is simple. Groups of order 20

(a) show how \(X\) may be regarded as a \(G\) -set in a natural way or, in other words, describe a natural group action of \(G\) on \(X ;\) (b) show that the action has the two properties required by the definition of an action; and (c) give a permutation representation of the action. \(G \approx \mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2} \cdot X=\) the vertices of a square.

In Exercises 3 through 8 describe the conjugacy classes and write the class equations of the indicated groups. $$ A_{4} $$

(a) find the stabilizer \(G_{a}\) for each \(a\) indicated, (b) find the orbit \(O_{a},(\mathrm{c})\) check the orbit-stabilizer relation \(\left|O_{a}\right|=\left[G: G_{a}\right],(\mathrm{d})\) determine whether the action is transitive, and (e) determine whether \(G\) acts faithfully on \(X\). \(X=\\{1,2,3,4\\},\) the four vertices of a square; \(G=\left\langle\rho^{2}, \tau\right\rangle,\) the subgroup generated by the rotation \(\rho^{2}\) of \(180^{\circ}\) and the flip \(\tau\) in \(D_{4} ; a=1,3\)

Let \(G=\mathbb{Z}\) and let \(X\) be the set of cosets of \(5 \mathbb{Z}\) in \(\mathbb{Z}\). Give an example of an action of \(G\) on \(X,\) defined in a natural way, that is not faithful.

In Exercises 3 through 8 describe the conjugacy classes and write the class equations of the indicated groups. $$ Z_{3} \times A_{4} $$

Let \(X=\mathrm{C}-\\{0,-1\\}\), the complex plane with 0 and -1 deleted. For \(z \in X\) let \(T_{0}(z)=z, T_{1}(z)=-1 /(1+z), T_{2}=(1+z) /-z,\) and let \(G=\left\\{T_{0}, T_{1}, T_{2}\right\\}\) (a) Show that \(G\) is a group under composition of functions. (b) Show how \(G\) acts on \(X\) in a natural way. (c) Find all \(a \in X\) such that \(G_{a}=G\)

In Exercises 1 through 11 find the number of essentially different ways in which we can do what is described. Replace each of the hydrogen atom in a molecule of benzene with a fluorine, chlorine, or bromine atom. (A molecule of benzene consists of six carbon atoms in a regular hexagon, with one hydrogen atom bonded to each.)

Show that no group of the indicated order is simple. Groups of order 75

Let \(G\) be any group. Show that for any \(a, b \in G,\) if \(a\) and \(b\) are conjugates, then they have the same order.

Show that if \(\tau=(x y)(u v)\) is the product of two disjoint 2 -cycles in \(S_{n},\) where \(n \geq 4,\) then the number of conjugates of \(\tau\) in \(S_{n}\) is \(n ! / 8 \cdot(n-4) !\)

For \(a, b \in \mathbb{R}\), let \(g(a, b) \in M(2, \mathbb{R})\) be the matrix $$ g(a, b)=\left[\begin{array}{ll} a & b \\ 0 & 1 \end{array}\right] $$ Let \(G=\\{g(a, b) \mid a, b \in \mathbb{R}, a \neq 0\\},\) with the operation of matrix multiplication, and let \(X=\mathbb{R},\) the real line. For \(g=g(a, b) \in G\) and \(x \in X,\) define \(g \cdot x=a x+b .\) (a) Show that \(G\) is a subgroup of \(G L(2, \mathbb{R})\). (b) Show that \(g \cdot x=a x+b\) defines an action of \(G\) on \(X\). (c) Find \(G_{0}\) and \(O_{0}\). (d) Does \(G\) act faithfully on \(X\) ? (e) Is this action transitive?