Chapter 15

Find all the Sylow 3-subgroups of \(S_{4}\) and show that they are all conjugate.

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

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

Prove that no group of order 96 is simple.

Prove that no group of order 160 is simple.

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 \(G\).

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.

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

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

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 \(G\).

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

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

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

Classify all the groups of order 175 up to isomorphism.

Show that every group of order 255 is cyclic.

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

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?

The 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 \(g P g^{-1}=h P h^{-1}\) Also, show that if \(N\) is the normalizer of \(P,\) then \(G=H N\).

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

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

Prove that a Sylow 2-subgroup of \(S_{5}\) is isomorphic to \(D_{4}\).

(a) Suppose \(p\) is prime and \(p\) does not divide \(m\). Show that $$ p \nmid\left(\begin{array}{c} p^{k} m \\ p^{k} \end{array}\right) . $$ (b) Let \(\mathcal{S}\) denote the set of all \(p^{k}\) element subsets of \(G\). Show that \(p\) does not divide \(|\mathcal{S}|\). (c) Define an action of \(G\) on \(\mathcal{S}\) by left multiplication, \(a T=\\{\) at \(: t \in T\\}\) for \(a \in G\) and \(T \in \mathcal{S}\). Prove that this is a group action. (d) Prove \(p \nmid\left|\mathcal{O}_{T}\right|\) for some \(T \in \mathcal{S}\). (e) Let \(\left\\{T_{1}, \ldots, T_{u}\right\\}\) be an orbit such that \(p \nmid u\) and \(H=\left\\{g \in G: g T_{1}=T_{1}\right\\}\). Prove that \(H\) is a subgroup of \(G\) and show that \(|G|=u|H|\) (f) Show that \(p^{k}\) divides \(|H|\) and \(p^{k} \leq|H|\). (g) Show that \(|H|=\left|\mathcal{O}_{T}\right| \leq p^{k} ;\) conclude that therefore \(p^{k}=|H| .\)

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