Chapter Questions
Let $H=\{(1),(12)(34),(13)(24),(14)(23)\} .$ Find the left cosets of $H$ in $A_{4}$ (see Table $5.1$ on page 111 ).
Let $H$ be as in Exercise $1 .$ How many left cosets of $H$ in $S_{4}$ are there? (Determine this without listing them.)
Let $H=\{0, \pm 3, \pm 6, \pm 9, \ldots\} .$ Find all the left cosets of $H$ in $Z$.
Rewrite the condition $a^{-1} b \in H$ given in property 5 of the lemma on page 145 in additive notation. Assume that the group is Abelian.
Let $H$ be as in Exercise 3 . Use Exercise 4 to decide whether or not the following cosets of $H$ are the same.a. $11+H$ and $17+H$b. $-1+H$ and $5+H$c. $7+H$ and $23+H$
Let $n$ be a positive integer. Let $H=\{0, \pm n, \pm 2 n, \pm 3 n, \ldots\} .$ Find all left cosets of $H$ in $Z$. How many are there?
Find all of the left cosets of $\{1,11\}$ in $U(30)$.
Suppose that $a$ has order 15 . Find all of the left cosets of $\left\langle a^{5}\right\rangle$ in $\langle a\rangle$.
Let $|a|=30$. How many left cosets of $\left\langle a^{4}\right\rangle$ in $\langle a\rangle$ are there? List them.
Give an example of a group $G$ and subgroups $H$ and $K$ such that $H K=\{h \in H, k \in K\}$ is not a subgroup of $G$.
If $H$ and $K$ are subgroups of $G$ and $g$ belongs to $G$, show that $g(H \cap K)=g H \cap g K$.
Let $a$ and $b$ be nonidentity elements of different orders in a group $G$ of order $155 .$ Prove that the only subgroup of $G$ that contains $a$ and $b$ is $G$ itself.
Let $H$ be a subgroup of $\mathbf{R}^{*}$, the group of nonzero real numbers under multiplication. If $\mathbf{R}^{+} \subseteq H \subseteq \mathbf{R}^{*}$, prove that $H=\mathbf{R}^{+}$ or $H=\mathbf{R}^{*}$.
Let $\mathbf{C}^{*}$ be the group of nonzero complex numbers under multiplication and let $H=\left\{a+b i \in \mathbf{C}^{*} \mid a^{2}+b^{2}=1\right\}$. Give a geometric description of the coset $(3+4 i) H$. Give a geometric description of the coset $(c+d i) H$.
Let $G$ be a group of order $60 .$ What are the possible orders for the subgroups of $G$ ?
Suppose that $K$ is a proper subgroup of $H$ and $H$ is a proper subgroup of $G$. If $|K|=42$ and $|G|=420$, what are the possible orders of $H$ ?
Let $G$ be a group with $|G|=p q$, where $p$ and $q$ are prime. Prove that every proper subgroup of $G$ is cyclic.
Recall that, for any integer $n$ greater than $1, \phi(n)$ denotes the number of positive integers less than $n$ and relatively prime to $n$. Prove that if $a$ is any integer relatively prime to $n$, then $a^{\phi(n)} \bmod n=1$.
Compute $5^{15} \bmod 7$ and $7^{13}$ mod 11 .
Use Corollary 2 of Lagrange's Theorem (Theorem $7.1$ ) to prove that the order of $U(n)$ is even when $n>2$.
Suppose $G$ is a finite group of order $n$ and $m$ is relatively prime to $n .$ If $g \in G$ and $g^{m}=e$, prove that $g=e$.
Suppose $H$ and $K$ are subgroups of a group $G .$ If $|H|=12$ and $|K|=35$, find $|H \cap K|$. Generalize.
Suppose that $H$ is a subgroup of $S_{4}$ and that $H$ contains (12) and (234). Prove that $H=S_{4}$.
Suppose that $H$ and $K$ are subgroups of $G$ and there are elements $a$ and $b$ in $G$ such that $a H \subseteq b K$. Prove that $H \subseteq K$.
Suppose that $G$ is an Abelian group with an odd number of elements. Show that the product of all of the elements of $G$ is the identity.
Suppose that $G$ is a group with more than one element and $G$ has no proper, nontrivial subgroups. Prove that $|G|$ is prime. (Do not assume at the outset that $G$ is finite.)
Let $|G|=15 .$ If $G$ has only one subgroup of order 3 and only one of order 5 , prove that $G$ is cyclic. Generalize to $|G|=p q$, where $p$ and $q$ are prime.
Let $G$ be a group of order $25 .$ Prove that $G$ is cyclic or $g^{5}=e$ for all $g$ in $G$. Generalize to any group of order $p^{2}$ where $p$ is prime. Does your proof work for this generalization?
Let $|G|=33$. What are the possible orders for the elements of $G$ ? Show that $G$ must have an element of order 3 .
Let $|G|=8$. Show that $G$ must have an element of order $2 .$
Can a group of order 55 have exactly 20 elements of order $11 ?$ Give a reason for your answer.
Determine all finite subgroups of $\mathbf{C}^{*}$, the group of nonzero complex numbers under multiplication.
Let $H$ and $K$ be subgroups of a finite group $G$ with $H \subseteq K \subseteq G$. Prove that $|G: H|=|G: K||K: H|$.
Suppose that a group contains elements of orders 1 through 10 . What is the minimum possible order of the group?
Give an example of the dihedral group of smallest order that contains a subgroup isomorphic to $Z_{12}$ and a subgroup isomorphic to $Z_{20}$. No need to prove anything, but explain your reasoning.
Show that in any group of order 100 , either every element has order that is a power of a prime or there is an element of order 10 .
Suppose that a finite Abelian group $G$ has at least three elements of order 3 . Prove that 9 divides $|G|$.
Prove that if $G$ is a finite group, the index of $Z(G)$ cannot be prime.
Find an example of a subgroup $H$ of a group $G$ and elements $a$ and $b$ in $G$ such that $a H \neq H b$ and $a H \cap H b \neq \phi .$ (Compare with property 5 of cosets.)
Prove that a group of order 63 must have an element of order 3 .
Let $G$ be a group of order 100 that has a subgroup $H$ of order 25 . Prove that every element of $G$ of order 5 is in $H$.
Let $G$ be a group of order $n$ and $k$ be any integer relatively prime to $n .$ Show that the mapping from $G$ to $G$ given by $g \rightarrow g^{k}$ is one-toone. If $G$ is also Abelian, show that the mapping given by $g \rightarrow g^{k}$ is an automorphism of $G$.
Let $G$ be a group of permutations of a set $S$. Prove that the orbits of the members of $S$ constitute a partition of $S .$ (This exercise is referred to in this chapter and in Chapter 29.)
Prove that every subgroup of $D_{n}$ of odd order is cyclic.
Let $G=\{(1),(12)(34),(1234)(56),(13)(24),(1432)(56),(56)(13),$,$(14)(23),(24)(56)\}$.a. Find the stabilizer of 1 and the orbit of 1 .b. Find the stabilizer of 3 and the orbit of 3 .c. Find the stabilizer of 5 and the orbit of 5 .
Prove that a group of order 12 must have an element of order $2 .$
Show that in a group $G$ of odd order, the equation $x^{2}=a$ has a unique solution for all $a$ in $G$.
Let $G$ be a group of order $p q r$, where $p, q$, and $r$ are distinct primes. If $H$ and $K$ are subgroups of $G$ with $|H|=p q$ and $|K|=q r$, prove that $|H \cap K|=q$.
Prove that a group that has more than one subgroup of order 5 must have order at least 25 .
Prove that $A_{5}$ has a subgroup of order 12 .
Prove that $A_{5}$ has no subgroup of order 30 .
Prove that $A_{5}$ has no subgroup of order 15 to $20 .$
Suppose that $\alpha$ is an element from a permutation group $G$ and one of its cycles in disjoint cycle form is $\left(a_{1} a_{2} \cdots a_{k}\right) .$ Show that $\left\{a_{1}\right.$, $\left.a_{2}, \ldots, a_{k}\right\} \subseteq$ orb $_{G}\left(a_{i}\right)$ for $1=1,2, \ldots, k$.
Let $G$ be a group and suppose that $H$ is a subgroup of $G$ with the property that for any $a$ in $G$ we have $a H=H a$. (That is, every element of the form $a h$ where $h$ is some element of $H$ can be written in the form $h_{1} a$ for some $h_{1} \in H .$ ) If $a$ has order 2, prove that the set $K=H \cup a H$ is a subgroup of $G$. Generalize to the case that $|a|=k$.
Prove that $A_{5}$ is the only subgroup of $S_{5}$ of order 60 .
Why does the fact that $A_{4}$ has no subgroup of order 6 imply that $\left|Z\left(A_{4}\right)\right|=1 ?$
Let $G=G L(2, \mathbf{R})$ and $H=S L(2, \mathbf{R})$. Let $A \in G$ and suppose that det $A=2$. Prove that $A H$ is the set of all $2 \times 2$ matrices in $G$ that have determinant $2 .$
Let $G$ be the group of rotations of a plane about a point $P$ in the plane. Thinking of $G$ as a group of permutations of the plane, describe the orbit of a point $Q$ in the plane. (This is the motivation for the name "orbit.")
Let $G$ be the rotation group of a cube. Label the faces of the cube 1 through 6 , and let $H$ be the subgroup of elements of $G$ that carry face 1 to itself. If $\sigma$ is a rotation that carries face 2 to face 1, give a physical description of the coset $H \sigma$.
The group $D_{4}$ acts as a group of permutations of the square regions shown below. (The axes of symmetry are drawn for reference purposes.) For each square region, locate the points in the orbit of the indicated point under $D_{4} .$ In each case, determine the stabilizer of the indicated point.
Let $G=G L(2, \mathbf{R})$, the group of $2 \times 2$ matrices over $\mathbf{R}$ with nonzero determinant. Let $H$ be the subgroup of matrices of determinant $\pm 1$. If $a, b \in G$ and $a H=b H$, what can be said about det $(a)$ and det $(b) ?$ Prove or disprove the converse. [Determinants have the property that det $(x y)=\operatorname{det}(x) \operatorname{det}(y) .]$
Calculate the orders of the following (refer to Figure $27.5$ for illustrations).a. The group of rotations of a regular tetrahedron (a solid with four congruent equilateral triangles as faces)b. The group of rotations of a regular octahedron (a solid with eight congruent equilateral triangles as faces)c. The group of rotations of a regular dodecahedron (a solid with 12 congruent regular pentagons as faces)d. The group of rotations of a regular icosahedron (a solid with 20 congruent equilateral triangles as faces)
Prove that the eight-element set in the proof of Theorem $7.5$ is a group.
A soccer ball has 20 faces that are regular hexagons and 12 faces that are regular pentagons. Use Theorem $7.4$ to explain why a soccer ball cannot have a $60^{\circ}$ rotational symmetry about a line through the centers of two opposite hexagonal faces.
If $G$ is a finite group with fewer than 100 elements and $G$ has subgroups of orders 10 and 25, what is the order of $G$ ?