Abstract Algebra: Theory and Applications

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 whose order is 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 $k<p\text{.}$ Show that $G$ must contain a normal subgroup.

14.

Let $H$ be a subgroup of a finite group $G\text{.}$ Prove that $gN(H)g^{-1}=N(gH{g}^{-1})$ 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 $P_1,\dots ,P_n$ 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?

21. The 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 $gP{g}^{-1}=hP{h}^{-1}\text{.}$ Also, show that if $N$ is the normalizer of $P\text{,}$ then $G=HN\text{.}$

22.

Show that if the order of $G$ is $p^{n}q\text{,}$ 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 $D_4\text{.}$

25. Another Proof of the Sylow Theorems.

1. Suppose $p$ is prime and $p$ does not divide $m\text{.}$ Show that

   $$p\nmid (\genfrac{}{}{0ex}{}{p^{k}m}{p^k})\text{.}$$

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 $\{T_1,\dots ,T_u\}$ be an orbit such that $p\nmid u$ and $H=\{g\in G:g{T}_1=T_1\}\text{.}$ Prove that $H$ is a subgroup of $G$ and show that $|G|=u|H|\text{.}$

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

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

26.

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