Chapter Questions
Show that conjugacy is an equivalence relation on a group.
If $a$ is a group element, prove that every element in $\operatorname{cl}(a)$ has the same order as $a$.
Let $a$ be a group element of even order. Prove that $a^{2}$ is not in $\operatorname{cl}(a)$.
Calculate all conjugacy classes for the quaternions (see Exercise 54 , Chapter 9).
Show that the function $T$ defined in the proof of Theorem $24.1$ is well-defined, is one-to-one, and maps the set of left cosets onto the conjugacy class of $a$.
Show that $\operatorname{cl}(a)=\{a\}$ if and only if $a \in Z(G)$.
Show that $Z_{2}$ is the only group that has exactly two conjugacy classes.
What can you say about the number of elements of order 7 in a group of order $168=8 \cdot 3 \cdot 7 ?$
Let $H$ be a subgroup of a group $G$. Prove that the number of conjugates of $H$ in $G$ is $|G: N(H)|$. (This exercise is referred to in this chapter.)
Let $H$ be a proper subgroup of a finite group $G$. Show that $G$ is not the union of all conjugates of $H$.
If $G$ is a group of odd order and $x \in G$, show that $x^{-1}$ is not in $\operatorname{cl}(x)$.
Determine the class equation for non-Abelian groups of orders 39 and 55 .
Determine which of the equations below could be the class equation given in the proof of Theorem $24.2 .$ For each part, provide your reasoning.a. $9=3+3+3$b. $21=1+1+3+3+3+3+7$c. $10=1+2+2+5$d. $18=1+3+6+8$
Exhibit a Sylow 2-subgroup of $S_{4}$. Describe an isomorphism from this group to $D_{4}$.
Suppose that $G$ is a group of order 48 . Show that the intersection of any two distinct Sylow 2 -subgroups of $G$ has order 8 .
Find all the Sylow 3-subgroups of $S_{4}$.
Let $K$ be a Sylow $p$ -subgroup of a finite group $G$. Prove that if $x \in$ $N(K)$ and the order of $x$ is a power of $p$, then $x \in K .$ (This exercise is referred to in this chapter.)
Suppose that $G$ is a group of order $p^{n} m$, where $p$ is prime and $p$ does not divide $m$. Show that the number of Sylow $p$ -subgroups divides $m$.
Suppose that $G$ is a group and $|G|=p^{n} m$, where $p$ is prime and $p>m$. Prove that a Sylow $p$ -subgroup of $G$ must be normal in $G$.
Let $H$ be a Sylow $p$ -subgroup of $G$. Prove that $H$ is the only Sylow $p$ -subgroup of $G$ contained in $N(H)$.
Suppose that $G$ is a group of order 168 . If $G$ has more than one Sylow 7-subgroup, exactly how many does it have?
Show that every group of order 56 has a proper nontrivial normal subgroup.
What is the smallest composite (that is, nonprime and greater than 1$)$ integer $n$ such that there is a unique group of order $n$ ?
Let $G$ be a noncyclic group of order 21 . How many Sylow 3subgroups does $G$ have?
Let $G$ be a group of order $p q$ where $p$ and $q$ are distinct primes and $p<q .$ Prove that the Sylow $q$ -subgroup is normal in $G$. (This exercise is referred to in this chapter.)
How many Sylow 5 -subgroups of $S_{5}$ are there? Exhibit two.
How many Sylow 3-subgroups of $S_{5}$ are there? Exhibit five.
What are the possibilities for the number of elements of order 5 in a group of order $100 ?$
What do the Sylow theorems tell you about any group of order 100 ?
Prove that a group of order 175 is Abelian.
Let $G$ be a group with $|G|=595=5 \cdot 7 \cdot 17$. Show that the Sylow 5-subgroup of $G$ is normal in $G$ and is contained in $Z(G)$.
Determine the number of Sylow 2 -subgroups of $D_{2 m}$, where $m$ is an odd integer at least 3 .
Generalize the argument given in Example 6 to obtain a theorem about groups of order $p^{2} q$, where $p$ and $q$ are distinct primes.
Prove that a group of order 375 has a subgroup of order 15 .
Without using Theorem $24.6$, prove that a group of order 15 is cyclic. (This exercise is referred to in the discussion about groups of order $30 .$ )
Prove that a group of order 105 contains a subgroup of order 35 .
Prove that a group of order 595 has a normal Sylow 17 -subgroup.
Let $G$ be a group of order $60 .$ Show that $G$ has exactly four elements of order 5 or exactly 24 elements of order $5 .$ Which of these cases holds for $A_{5}$ ?
Show that the center of a group of order 60 cannot have order $4 .$
Suppose that $G$ is a group of order 60 and $G$ has a normal subgroup $N$ of order $2 .$ Show thata. $G$ has normal subgroups of orders 6,10 , and 30 .b. $G$ has subgroups of orders 12 and 20 .c. $G$ has a cyclic subgroup of order 30 .
Let $G$ be a group of order 60 . If the Sylow 3-subgroup is normal, show that the Sylow 5 -subgroup is normal.
Show that if $G$ is a group of order 168 that has a normal subgroup of order 4, then $G$ has a normal subgroup of order 28 .
Suppose that $p$ is prime and $|G|=p^{n} .$ Show that $G$ has normal subgroups of order $p^{k}$ for all $k$ between 1 and $n$ (inclusive).
Suppose that $G$ is a group of order $p^{n}$, where $p$ is prime, and $G$ has exactly one subgroup for each divisor of $p^{n}$. Show that $G$ is cyclic.
Suppose that $p$ is prime and $|G|=p^{n}$. If $H$ is a proper subgroup of $G$ prove that $N(H)>H$. (This exercise is referred to in Chapter 25.)
If $H$ is a finite subgroup of a group $G$ and $x \in G$, prove that $|N(H)|=\left|N\left(x H x^{-1}\right)\right|$
Let $H$ be a Sylow 3 -subgroup of a finite group $G$ and let $K$ be a Sylow 5-subgroup of $G$. If 3 divides $|N(K)|$, prove that 5 divides $|N(H)|$.
If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^{k}$ for some prime $p$, show that $H$ is contained in every Sylow $p$ -subgroup of $G$.
Suppose that $G$ is a finite group and $G$ has a unique Sylow $p$ -subgroup for each prime $p$. Prove that $G$ is the internal direct product of its nontrivial Sylow $p$ -subgroups. If each Sylow $p$ -subgroup is cyclic, is $G$ cyclic? If each Sylow $p$ -subgroup is Abelian, is $G$ Abelian?
Suppose that $G$ is a finite group and $G$ has exactly one subgroup for each divisor of $|G|$. Prove that $G$ is cyclic.
Let $G$ be a finite group and let $H$ be a normal Sylow $p$ -subgroup of $G$. Show that $\alpha(H)=H$ for all automorphisms $\alpha$ of $G$.
If $H$ is a Sylow $p$ -subgroup of a group, prove that $N(N(H))=N(H)$.
Let $p$ be a prime and $H$ and $K$ be Sylow $p$ -subgroups of a group $G$. Prove that $|N(H)|=|N(K)|$.
Let $G$ be a group of order $p^{2} q^{2}$, where $p$ and $q$ are distinct primes, $q \times p^{2}-1$, and $p+q^{2}-1$. Prove that $G$ is Abelian. List three pairs of primes that satisfy these conditions.
Let $H$ be a normal subgroup of a group $G$. Show that $H$ is the union of the conjugacy classes in $G$ of the elements of $H$. Is this true when $H$ is not normal in $G ?$
Let $G$ be a finite group and $p$ be a prime that divides $|G|$. If $H$ is a Sylow $p$ -subgroup of $N(H)$, prove that $H$ is a Sylow $p$ -subgroup of $G$.
Show that a group of order 12 cannot have nine elements of order 2 .
If $|G|=36$ and $G$ is non-Abelian, prove that $G$ has more than one Sylow 2-subgroup or more than one Sylow 3-subgroup.
Let $G$ be a non-Abelian group of order $p q$ where $p$ and $q$ are primes and $p<q$. Prove that $G$ has exactly $q+1$ nontrivial proper subgroups.
Determine the groups of order 45 .
Explain why a group of order $4 m$ where $m$ is odd must have a subgroup isomorphic to $Z_{4}$ or $Z_{2} \oplus Z_{2}$ but cannot have both a subgroup isomorphic to $Z_{4}$ and a subgroup isomorphic to $Z_{2} \oplus Z_{2}$. Show that $S_{4}$ has a subgroup isomorphic to $Z_{4}$ and a subgroup isomorphic to $Z_{2} \oplus Z_{2}$
Let $p$ be the smallest prime that divides the order of a finite group $G$. If $H$ is a Sylow $p$ -subgroup of $G$ and is cyclic, prove that $N(H)=C(H)$.
Let $G$ be a group of order $715=5 \cdot 11 \cdot 13 .$ Let $H$ be a Sylow 13-subgroup of $G$ and $K$ be a Sylow 11 -subgroup of $G$. Prove that $H$ is contained in $Z(G)$. Can the argument you used to prove that $H$ is contained in $Z(G)$ also be used to show that $K$ is contained in $Z(G)$ ?