Exercises 10.3 Exercises

5.

Let \(G = A_4\) and suppose that \(G\) acts on itself by conjugation; that is, \((g,h)~\mapsto~ghg^{-1}\text{.}\)

Determine the conjugacy classes (orbits) of each element of \(G\text{.}\)
Determine all of the isotropy subgroups for each element of \(G\text{.}\)

6.

Find the conjugacy classes and the class equation for each of the following groups.

\(\displaystyle S_4\)
\(\displaystyle D_5\)
\(\displaystyle {\mathbb Z}_9\)
\(\displaystyle Q_8\)

Hint.

The conjugacy classes for \(S_4\) are

\begin{gather*} {\mathcal O}_{(1)} = \{ (1) \},\\ {\mathcal O}_{(12)} = \{ (1 \, 2), (1 \, 3), (1 \, 4), (2 \, 3), (2 \, 4), (3 \, 4) \},\\ {\mathcal O}_{(1 \, 2)(3 \, 4)} = \{ (1 \, 2)(3 \, 4), (1 \, 3)(2 \, 4), (1 \, 4)(2 \, 3) \},\\ {\mathcal O}_{(123)} = \{ (1 \, 2 \, 3), (1 \, 3 \, 2), (1 \, 2 \, 4), (1 \, 4 \, 2), (1 \, 3 \, 4), (1 \, 4 \, 3), (2 \, 3 \, 4), (2 \, 4 \, 3) \},\\ {\mathcal O}_{(1234)} = \{ (1 \, 2 \, 3 \, 4), (1 \, 2 \, 4 \, 3), (1 \, 3 \, 2 \, 4), (1 \, 3 \, 4 \, 2), (1 \, 4 \, 2 \, 3), (1 \, 4 \, 3 \, 2) \}\text{.} \end{gather*}

The class equation is \(1 + 3 + 6 + 6 + 8 = 24\text{.}\)

7.

Write the class equation for \(S_5\) and for \(A_5\text{.}\)

8.

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

Hint.

If \(|G| = 18 = 2 \cdot 3^2\text{,}\) then the order of a Sylow \(2\)-subgroup is \(2\text{,}\) and the order of a Sylow \(3\)-subgroup is \(9\text{.}\)

9.

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

Hint.

The four Sylow \(3\)-subgroups of \(S_4\) are \(P_1 = \{ (1), (1 \, 2 \, 3), (1 \, 3 \, 2) \}\text{,}\) \(P_2 = \{ (1), (1 \, 2 \, 4), (1 \, 4 \, 2) \}\text{,}\) \(P_3 = \{ (1), (1 \, 3 \, 4), (1 \, 4 \, 3) \}\text{,}\) \(P_4 = \{ (1), (2 \, 3 \, 4), (2 \, 4 \, 3) \}\text{.}\)

10.

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

11.

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

12.

Prove that no group of order \(96\) is simple.

Hint.

Since \(|G| = 96 = 2^5 \cdot 3\text{,}\) \(G\) has either one or three Sylow \(2\)-subgroups by the Third Sylow Theorem. If there is only one subgroup, we are done. If there are three Sylow \(2\)-subgroups, let \(H\) and \(K\) be two of them. Therefore, \(|H \cap K| \geq 16\text{;}\) otherwise, \(HK\) would have \((32 \cdot 32)/8 = 128\) elements, which is impossible. Thus, \(H \cap K\) is normal in both \(H\) and \(K\) since it has index \(2\) in both groups.

13.

Prove that no group of order \(160\) is simple.

14.

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 \(G\text{.}\)

15.

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.

Hint.

Show that \(G\) has a normal Sylow \(p\)-subgroup of order \(p^2\) and a normal Sylow \(q\)-subgroup of order \(q^2\text{.}\)

16.

Show that a group of order \(33\) has only one Sylow \(3\)-subgroup.

17.

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

Hint.

False.

18.

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 \(G\text{.}\)

19.

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

20.

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

21.

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

22.

Prove that a group of order \(108\) must have a proper nontrivial normal subgroup.

23.

Classify all the groups of order \(175\) up to isomorphism.

24.

Show that every group of order \(255\) is cyclic.

Hint.

If \(G\) is abelian, then \(G\) is cyclic, since \(|G| = 3 \cdot 5 \cdot 17\text{.}\)

25.

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 \(P_1 \times \cdots \times P_n\text{.}\)

26.

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

27.

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?

28. 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 \(G= HN\text{.}\)

29.

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 proper nontrivial normal subgroup.

30.

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

Hint.

Define a mapping between the right cosets of \(N(H)\) in \(G\) and the conjugates of \(H\) in \(G\) by \(N(H) g \mapsto g^{-1} H g\text{.}\) Prove that this map is a bijection.

31.

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

32. Another Proof of the Sylow Theorems.

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*}
Let \({\mathcal S}\) denote the set of all \(p^k\) element subsets of \(G\text{.}\) Show that \(p\) does not divide \(|{\mathcal S}|\text{.}\)
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.
Prove \(p \nmid | {\mathcal O}_T|\) for some \(T \in {\mathcal S}\text{.}\)
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 \(|G| = u |H|\text{.}\)
Show that \(p^k\) divides \(|H|\) and \(p^k \leq |H|\text{.}\)
Show that \(|H| = |{\mathcal O}_T| \leq p^k\text{;}\) conclude that therefore \(p^k = |H|\text{.}\)

33.

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.

Hint.

Let \(a G', b G' \in G/G'\text{.}\) Then \((a G')( b G') = ab G' = ab(b^{-1}a^{-1}ba) G' = (abb^{-1}a^{-1})ba G' = ba G'\text{.}\)

34.

A group acts faithfully on a \(G\)-set \(X\) if the identity is the only element of \(G\) that leaves every element of \(X\) fixed. Show that \(G\) acts faithfully on \(X\) if and only if no two distinct elements of \(G\) have the same action on each element of \(X\text{.}\)

35.

Let \(p\) be prime. Show that the number of different abelian groups of order \(p^n\) (up to isomorphism) is the same as the number of conjugacy classes in \(S_n\text{.}\)

36.

Let \(a \in G\text{.}\) Show that for any \(g \in G\text{,}\) \(gC(a) g^{-1} = C(gag^{-1})\text{.}\)

Hint.

Use the fact that \(x \in g C(a) g^{-1}\) if and only if \(g^{-1}x g \in C(a)\text{.}\)

37.

Let \(|G| = p^n\) be a nonabelian group for \(p\) prime. Prove that \(|Z(G)| \lt p^{n - 1}\text{.}\)

38.

Let \(G\) be a group with order \(p^n\) where \(p\) is prime and \(X\) a finite \(G\)-set. If \(X_G = \{ x \in X : gx = x \text{ for all }g \in G \}\) is the set of elements in \(X\) fixed by the group action, then prove that \(|X| \equiv |X_G| \pmod{ p}\text{.}\)

39.

If \(G\) is a group of order \(p^n\text{,}\) where \(p\) is prime and \(n \geq 2\text{,}\) show that \(G\) must have a proper subgroup of order \(p\text{.}\) If \(n \geq 3\text{,}\) is it true that \(G\) will have a proper subgroup of order \(p^2\text{?}\)

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