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## Section15.3Exercises

###### 1

What are the orders of all Sylow $p$-subgroups where $G$ has order $18\text{,}$ $24\text{,}$ $54\text{,}$ $72\text{,}$ and $80\text{?}$

###### 2

Find all the Sylow $3$-subgroups of $S_4$ and show that they are all conjugate.

###### 3

Show that every group of order $45$ has a normal subgroup of order $9\text{.}$

###### 4

Let $H$ be a Sylow $p$-subgroup of $G\text{.}$ Prove that $H$ is the only Sylow $p$-subgroup of $G$ contained in $N(H)\text{.}$

###### 5

Prove that no group of order $96$ is simple.

###### 6

Prove that no group of order $160$ is simple.

###### 7

If $H$ is a normal subgroup of a finite group $G$ and $|H| = p^k$ for some prime $p\text{,}$ show that $H$ is contained in every Sylow $p$-subgroup of $G\text{.}$

###### 8

Let $G$ be a group of order $p^2 q^2\text{,}$ where $p$ and $q$ are distinct primes such that $q \nmid p^2 - 1$ and $p \nmid q^2 - 1\text{.}$ Prove that $G$ must be abelian. Find a pair of primes for which this is true.

###### 9

Show that a group of order $33$ has only one Sylow $3$-subgroup.

###### 10

Let $H$ be a subgroup of a group $G\text{.}$ Prove or disprove that the normalizer of $H$ is normal in $G\text{.}$

###### 11

Let $G$ be a finite group divisible by a prime $p\text{.}$ Prove that if there is only one Sylow $p$-subgroup in $G\text{,}$ it must be a normal subgroup of $G\text{.}$

###### 12

Let $G$ be a group of order $p^r\text{,}$ $p$ prime. Prove that $G$ contains a normal subgroup of order $p^{r-1}\text{.}$

###### 13

Suppose that $G$ is a finite group of order $p^n k\text{,}$ where $k \lt p\text{.}$ Show that $G$ must contain a normal subgroup.

###### 14

Let $H$ be a subgroup of a finite group $G\text{.}$ Prove that $g N(H) g^{-1} = N(gHg^{-1})$ for any $g \in G\text{.}$

###### 15

Prove that a group of order $108$ must have a normal subgroup.

###### 16

Classify all the groups of order $175$ up to isomorphism.

###### 17

Show that every group of order $255$ is cyclic.

###### 18

Let $G$ have order $p_1^{e_1} \cdots p_n^{e_n}$ and suppose that $G$ has $n$ Sylow $p$-subgroups $P_1, \ldots, P_n$ where $|P_i| = p_i^{e_i}\text{.}$ Prove that $G$ is isomorphic to $P_1 \times \cdots \times P_n\text{.}$

###### 19

Let $P$ be a normal Sylow $p$-subgroup of $G\text{.}$ Prove that every inner automorphism of $G$ fixes $P\text{.}$

###### 20

What is the smallest possible order of a group $G$ such that $G$ is nonabelian and $|G|$ is odd? Can you find such a group?

###### 21The Frattini Lemma

If $H$ is a normal subgroup of a finite group $G$ and $P$ is a Sylow $p$-subgroup of $H\text{,}$ for each $g \in G$ show that there is an $h$ in $H$ such that $gPg^{-1} = hPh^{-1}\text{.}$ Also, show that if $N$ is the normalizer of $P\text{,}$ then $G= HN\text{.}$

###### 22

Show that if the order of $G$ is $p^nq\text{,}$ where $p$ and $q$ are primes and $p>q\text{,}$ then $G$ contains a normal subgroup.

###### 23

Prove that the number of distinct conjugates of a subgroup $H$ of a finite group $G$ is $[G : N(H) ]\text{.}$

###### 24

Prove that a Sylow $2$-subgroup of $S_5$ is isomorphic to $D_4\text{.}$

###### 25Another Proof of the Sylow Theorems
1. Suppose $p$ is prime and $p$ does not divide $m\text{.}$ Show that

\begin{equation*} p \nmid \binom{p^k m}{p^k}. \end{equation*}
2. Let ${\mathcal S}$ denote the set of all $p^k$ element subsets of $G\text{.}$ Show that $p$ does not divide $|{\mathcal S}|\text{.}$

3. Define an action of $G$ on ${\mathcal S}$ by left multiplication, $aT = \{ at : t \in T \}$ for $a \in G$ and $T \in {\mathcal S}\text{.}$ Prove that this is a group action.

4. Prove $p \nmid | {\mathcal O}_T|$ for some $T \in {\mathcal S}\text{.}$

5. Let $\{ T_1, \ldots, T_u \}$ be an orbit such that $p \nmid u$ and $H = \{ g \in G : gT_1 = T_1 \}\text{.}$ Prove that $H$ is a subgroup of $G$ and show that $|G| = u |H|\text{.}$

6. Show that $p^k$ divides $|H|$ and $p^k \leq |H|\text{.}$

7. Show that $|H| = |{\mathcal O}_T| \leq p^k\text{;}$ conclude that therefore $p^k = |H|\text{.}$

###### 26

Let $G$ be a group. Prove that $G' = \langle a b a^{-1} b^{-1} : a, b \in G \rangle$ is a normal subgroup of $G$ and $G/G'$ is abelian. Find an example to show that $\{ a b a^{-1} b^{-1} : a, b \in G \}$ is not necessarily a group.