Exercises15.4Exercises

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 $$k \lt p\text{.}$$ 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?

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 $$gPg^{-1} = hPh^{-1}\text{.}$$ 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 $$D_4\text{.}$$

25.Another 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}\text{.} \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.