Sylow Theorems

Show that conjugacy is an equivalence relation on a group.

If $a$ is a group element, prove that every element in $cl(a)$ has the same order as $a$.

Let $a$ be a group element of even order. Prove that $a^2$ is not in $cl(a)$.

Calculate all conjugacy classes for the quaternions (see Exercise 54, Chapter 9).

Show that the function $T$ defined in the proof of Theorem 23.1 is well-defined, is one-to-one, and maps the set of left cosets onto the conjugacy class of $a$.

(Theorem 23.1, Number of Conjugates of $a$: Let $G$ be a finite group and let $a$ be an element of $G$. Then $|cl(a)| = |G : C(a)|$, where $C(a)$ is the centralizer of $a$. In the proof, $T$ is the function that sends the coset $xC(a)$ to the conjugate $xax^{-1}$ of $a$.)

Show that $cl(a) = \left\{ a\right\}$ if and only if $a \in Z(G)$.

Show that $\mathbb {Z}_2$ 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 $cl(x)$.

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 23.2 (which shows that a nontrivial finite group of prime-power order has a nontrivial center). For each part, provide your reasoning.

Determine the class equation for $D_6$.

Suppose that $G$ is a group of order $48$. Show that the intersection of any two distinct Sylow 2-subgroups of $G$ has order $8$.

If $a$ and $b$ are group elements and $i$ and $k$ are positive integers, prove that $bab^{-1} = a^i$ implies that $b^k a b^{-k} = a^{i^k}$.

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 the form $\left\{ a^ib^j \mid i = 0, 1, \ldots , 6,\ j = 0, 1, 2\right\}$ and $ba = ab^2$. Express $aba^2b$ in the form $a^ib^j$.

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 (i.e., 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 3-subgroups does $G$ have?

Let $G$ be a group of order $pq$ 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.)

Prove that none of $S_5, S_6, S_7, A_5, A_6$, and $A_7$ can have a normal subgroup of order $3$ or $5$.

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 $Z(G)$.

One can construct a non-Abelian group of order $21$ of the form $\left\{ a^ib^j\right\}$ where $|a| = 7$ and $|b| = 3$ and multiplication is defined using either the relation $ba = a^2b$ or the relation $ba = a^4b$. Use this information to find relationships among $|a|$, $|b|$, $2$ and $4$.

One can construct a non-Abelian group of order $105$ of the form $a^ib^j$ where $|a| = 35$ and $|b| = 3$ and multiplication is defined using either the relation $ba = a^{11}b$ or the relation $ba = a^{18}b$. Use this information to find relationships among $|a|$, $|b|$, $11$, and $16$.

Explain why there cannot exist a group $G$ such that $[G : Z(G)] = 15$.

Let $G$ be a finite group with more than one element and let $H$ be the intersection of all non-trivial subgroups of $G$. What can you say about $|H|$?

Generalize the argument given in Example 6 to obtain a theorem about groups of order $p^2q$, where $p$ and $q$ are distinct primes.

Prove that a group of order $375$ has a subgroup of order $15$.

Without using Theorem 23.6, 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 normal 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 that

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 24.)

If $H$ is a finite subgroup of a group $G$ and $x \in G$, prove that $|N(H)| = |N(xHx^{-1})|$.

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 $|N(H)|$.

Prove that a group of order $48$ has a normal subgroup of order $16$ or a normal subgroup of order $8$.

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 $p^2q^2$, where $p$ and $q$ are distinct primes, $q \nmid p^2 - 1$, and $p \nmid q^2 - 1$. 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 $pq$ 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 $4m$ where $m$ is odd must have a subgroup isomorphic to $\mathbb {Z}_4$ or $\mathbb {Z}_2 \oplus \mathbb {Z}_2$ but cannot have both a subgroup isomorphic to $\mathbb {Z}_4$ and a subgroup isomorphic to $\mathbb {Z}_2 \oplus \mathbb {Z}_2$. Show that $S_4$ has a subgroup isomorphic to $\mathbb {Z}_4$ and a subgroup isomorphic to $\mathbb {Z}_2 \oplus \mathbb {Z}_2$.

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 $Z(G)$?

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 $G$.