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 \(9\text{.}\)
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.
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 \(G\text{.}\)
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.
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 \(G\text{.}\)
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.
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 \(P_1 \times \cdots \times P_n\text{.}\)
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 \(G= HN\text{.}\)
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.
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.
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.
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 \(|G| = u |H|\text{.}\)
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.
