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Section15.3Exercises

1

What are the orders of all Sylow \(p\)-subgroups where \(G\) has order 18, 24, 54, 72, and 80?

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.

4

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

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 \(p\), show that \(H\) is contained in every Sylow \(p\)-subgroup of \(G\).

8

Let \(G\) be a group of order \(p^2 q^2\), where \(p\) and \(q\) are distinct primes such that \(q \nmid p^2 - 1\) and \(p \nmid q^2 - 1\). 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 \(G\). Prove or disprove that the normalizer of \(H\) is normal in \(G\).

11

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

12

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

13

Suppose that \(G\) is a finite group of order \(p^n k\), where \(k \lt p\). Show that \(G\) must contain a normal subgroup.

14

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

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 \(|P_i| = p_i^{e_i}\). Prove that \(G\) is isomorphic to \(P_1 \times \cdots \times P_n\).

19

Let \(P\) be a normal Sylow \(p\)-subgroup of \(G\). Prove that every inner automorphism of \(G\) fixes \(P\).

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 \(H\), for each \(g \in G\) show that there is an \(h\) in \(H\) such that \(gPg^{-1} = hPh^{-1}\). Also, show that if \(N\) is the normalizer of \(P\), then \(G= HN\).

22

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

23

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

24

Prove that a Sylow 2-subgroup of \(S_5\) is isomorphic to \(D_4\).

25Another Proof of the Sylow Theorems

  1. Suppose \(p\) is prime and \(p\) does not divide \(m\). 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 \(G\). Show that \(p\) does not divide \(|{\mathcal S}|\).

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

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

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

  6. Show that \(p^k\) divides \(|H|\) and \(p^k \leq |H|\).

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

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.